Guide to Preambles in ETC

Prepared by Clark Kimberling (August 2026)

This Guide catalogs the inter-center group preambles across all 37 Parts of the Encyclopedia of Triangle Centers (ETC), providing a searchable index of thematic center families, geometric transformations, contributors, and definitions.

Part 1

• X₂–X₃₀ — Points on the Euler line

• X₇₄–X₁₁₂ — Points on the circumcircle; mappings Λ and Ψ

• X₁₁₃–X₁₃₉ — Points on the nine-point circle

• X₁₄₀–X₁₇₀ — Centers of central triangles

• X₁₉₁–X₂₃₆ — Ceva conjugates

• X₂₃₇–X₂₄₈ — Line conjugates

• X₂₄₉–X₂₉₇ — isogonal conjugates of prevously listed centers

• X₂₉₈–X₃₅₀ — isotomic conjugates of prevously listed centers

• X₄₀₁–X₄₇₅ — on the Euler line

• X₄₈₅–X₄₉₅ — triangle centers associated with squares

• X₅₀₃–X₅₁₀ — Isoscelizer points: X₅₀₃-X₅₁₀

• X₅₁₁–X₅₂₆ — Centers on the lines at infinity: X₅₁₁-X₅₂₆

• X₅₂₇–X₅₆₅

  Among these, X₅₂₇-X₅₄₅ are of the form “direction of vector AX+BY+CX for selected triangle centers X. Points X₅₄₆-X₅₅₁ are midpoints.

• X₅₆₆–X₅₈₄

  (includes definitions of these words as nouns: orthogonal, harmonic, orthoharmonic.)

• X₆₁₆–X₆₄₂

  contributed by Bernard Gibert, March 2, 2001. Trilinears for points in this section were found by Joe Goggins, Oct. 19, 2005.

Part 2

• X₁₁₁₅–X₁₁₅₀

• X₁₁₆₆–X₁₂₀₈ — Saragossa points (1st, 2nd, 3rd)

  contributed by Darij Grinberg

• X₁₂₈₆–X₁₃₁₁ — Collings Transforms

• X₁₃₅₄–X₁₃₆₇ — Brisse Transforms

• X₁₃₉₃–X₁₄₇₇ — Beth Conjugates

• X₁₅₁₂–X₁₅₆₈ — Orthojoins

• X₁₆₀₁–X₁₆₃₄ — TCC Perspectors

  Suppose P is a point. As noted in TCCT, p. 201, the tangential triangle is perspective to the circumcevian triangle of P. (The tangential triangle is also perspector to the circum-anticevian triangle of P, with the same perspector as for the circumcevian triangle of P.) In August 2003, Jean-Pierre Ehrmann gave barycentrics for the perspector, called the TCC-perspector of P. For further properties of TCC perpsectors, published some 14 years after their introduction here, see I. Minevich and P. Morton, International Journal of Geometry 2017, “Synthetic foundations of cevian geometry, IV”

• X₁₆₃₅–X₁₆₅₁ — Tripolar Centroids

  contributed by Darij Grinberg, August 24, 2003.

• X₁₆₆₂–X₁₇₀₆ — Circle-related points

  contributed by Peter Moses during August, 2003.

• X₁₇₀₇–X₁₇₈₈ — Mimosa transforms and inverse Mimosa tranforms

  contributed by Clark Kimberling, September 16, 2003. Let g(P,X) denote the P-gimel conjugate of X. The Mimosa transform M(X) arises in connection with the equation \(g(P,X) = X\).

• X₁₈₂₄–X₁₉₀₇ — Zozma transforms

  (isogonal conjugates of inverse Mimosa transforms)

• X₁₉₀₈–X₁₉₈₂ — Centers from bicentric pairs

  Suppose P and U are a bicentric pair. Many operations on P and U result in triangle centers. Among these are trilinear and barycentric product, bicentric sum, bicentric difference, crosssum, and crossdifference.

• X₁₉₉₂–X₂₀₀₆ — Orthocorrespondents

  Suppose that P is a point in the plane of triangle ABC. The perpendiculars through P to the lines AP, BP, CP meet the lines BC, CA, AB, respectively, in collinear points. Let L denote their line. The trilinear pole of L is the orthocorrespondent of :g:`P <P*>`. This definition was introduced by Bernard Gibert.

• X₂₀₀₇–X₂₀₄₀ — Gallatly circle

  (pedal cicle of the 1st and 2nd Brocard points).

• X₂₀₄₁–X₂₀₄₆ — Euler-Vecten-Gibert Points

  On August 13, 2003, Bernard Gibert contributed six centers that lie on the Euler line and are related to the Vecten points.

• X₂₀₅₅–X₂₀₆₅ — Orion transforms

  contributed by Jean-Peirre Ehramann, September 24, 2003.

• X₂₀₇₀–X₂₀₈₀ — Inverses in circumcircle

  contributed by Peter Moses.

• X₂₀₈₁–X₂₀₈₈ — PK and NK Transforms

  contributed by Bernard Gibert, October 1, 2003. Let \(X ^{-1}\) denote the isogonal conjugate of X. Then PK(X) is the point of intersection of the trilinear polar of X and the trilinear polar of \(X ^{-1}\), and NK(P) is the pole of the line \(XX^{-1}\) with respect to the conic that passes through points A, B, C, X, and \(X ^{-1}\). Also, PK(X) is the crossdifference, and NK(X) the crosssum, of X and \(X ^{-1}\).

• X₂₀₉₃–X₂₁₀₅ — Reflections

  contributed by Peter Moses.

• X₂₁₀₆–X₂₁₁₉ — points on the 2nd equal-areas cubic, EAC2

  For any point P on EAC2, the X₂-isoconjugate of P is also on EAC2. Contributed by Clark Kimberling.

• X₂₁₂₀–X₂₁₄₃ — Eigencenters and eigentransforms

  Contributed by Clark Kimberling, October, 2003.

• X₂₃₆₅–X₂₃₈₄ — More points on the circumcircle

  October 20, 2003, revised January 20, 2015 following suggestions by Viktor Kataysky. Notations: CIR(U)

• X₂₃₉₄–X₂₄₁₉ — Gibert-Simson Transforms

  On October 19, 2003, Bernard Gibert contributed points that lie on the Simson cubic. Indeed, this cubic is the image of the circumcircle under a mapping here named the Gibert-Simson transform.

• X₂₄₄₆–X₂₅₇₃ — (Mostly) circle-related points

  contributed by Peter J. C. Moses, October-November, 2003. See the notes just before X₁₆₆₂ for an introduction and notation. Included in this section are centers of similitude of several pairs of circles.

• X₂₅₉₄–X₂₆₇₀ — related to bicentric pairs

• X₂₆₇₇–X₂₇₇₀ — Rigby-Simson points and Simson-Rigby points

• X₂₈₅₅–X₂₈₆₈ — Simson-Moses points

  defined by changing “isogonal” to “isotomic” in the definition of Simson-Ribgy points (X₂₆₈₇-X₂₇₇₀.

• X₂₈₈₃–X₂₉₆₂ — Isogonal Conjugates with respect to Special Triangles

  (e.g,, medial, orthic, excentral, tangential). The term “complementary conjugate” already in use at the time of this writing (November 11, 2004) is a synonym for “medial isogonal conjugate”, as is “anticomplementary conjugate” for “anticomplementary isogonal conjugate”; accordingly, the earlier terms will be used in the sequel. Also, “excentral isogonal conjugate” is “X₁₈₈-aleph conjugate” and “orthic isogonal conjugate” is “X₄-Ceva conjugate”; in these cases, the new terminology will be used.

• X₂₉₆₇–X₂₉₇₃ — Points lying on the MacBeath inconic

  contributed by Peter Moses, November 12, 2004,

• X₂₉₇₉–X₂₉₉₈ — Dual triangles: DC points and CD points

  contributed by Clark Kimberling. Suppose DEF is a triangle in the plane of triangle ABC. Let D' be the isogonal conjugate of the point of intersection of line EF and the line at infinity. Define E' and F' cyclically. The triangle D'E'F' is here named the dual of DEF. The vertices of D'E'F' lie on the circumcircle, and D'E'F' is similar to DEF. The duality is between the sidelines EF, FD, DE and the points D', E', F', respectively. For example, if E and F remain fixed and D varies, then D' remains fixed, while E'F' varies. Actually D'E'F' is the dual of any triangle homothetic to DEF. Indeed, DEF need not be a triangle but can be the union of three concurrent lines. Suppose U is a point having cevian triangle DEF and dual triangle D'E'F'. Then there exists a point DC(U) whose circum-anticevian triangle (TCCT, p. 201) is D'E'F'. To construct DC(U) from U, let A' = AD'∩BC, and let A″ be the {B,C}-harmonic conjugate of A'. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in DC(U). Also, DC(U) = U-isoconjugate of the crosssum of U and X₆.

• X₃₀₀₀–X₃₀₁₉ — Intersections of Central Lines

Part 3

• X₃₀₂₇–X₃₀₂₈ — Antipodal Pairs on Circles

  In response to Stevanovic’s findings (X₃₀₀₀ to X₃₀₀₆), Peter Moses noted (Hyacinthos, 12/9/2004) a method for mapping a pair of antipodal points on one circle to an antipodal pair on another circle. The method depends on centers of similitude: Suppose O₁ and O₂ are circles, that P is on O₁ and that P' is the antipode of P on O₁. Let U be the internal center of similitude (insimilicenter) of O₁ and O₂, and V the exsimilicenter. Define Q = PU ∩P' V and Q' = PV ∩P' U. Then on O₂, point Q' is the antipode of Q. Moreover, the lines PP' and QQ' are parallel.

• X₃₁₁₂–X₃₁₁₈

  represent points introduced by Quang Tuan Bui in Hyacinthos, August 5, 2006. Suppose X is a point not on a sideline of ABC. Let

  gX = isogonal conjugate of X

  tX = isotomic conjugate of X

  tgX = isotomic conjugate of gX

  gtX = isogonal conjugate of tX

  Gt = intersection of lines X(tX) and (gX)(gtX)

  Tg = intersection of lines X(gX) and (tX)(tgX)

  Then the points A, B, C, gX, tX, Tg, Gt are on a conic. As a circumconic, it is the image under the isogonal conjugate mapping of line X(gtX). It is also the image under the isotomic conjugate mapping of line X(tgX).

• X₃₁₁₉–X₃₁₂₆ — Danneels Perspectors

  Suppose A₁B₁C₁ is the cevian triangle of a point X. Let L_AB be the line through B parallel to A₁B₁, and let L_AC be the line through C parallel to A₁C₁.

  Let A₂ = L_AB∩L_AC, and define B₂ and C₂ cyclically. Let

  A₃ = BB₂∩CC₂, B₃ = CC₂∩AA₂, C₃ = AA₂∩BB₂.

  Eric Danneels proves in “A Simple Perspectivity,” :cite:`Forum Geometricorum <forum-geometricorum>` 6 (2006) 199-203, that the triangles A₃B₃C₃ and ABC are perspective. If X =\(x : y : z\) (barycentrics), then the Danneels perspector P(X) is given by

  P(X) = x(y - z)² : y(z - x)² : z(x - y)².

• X₃₂₇₂–X₃₂₈₃ — are associated with six equilateral triangles

  related to the (1st) Morley triangle; trilinears for vertices of these triangles are given just below. Received from Milorad R. Stevanovic, November 24, 2007 and December 25, 2007.

• X₃₃₁₀ — Barycentric Products of Perpendicular Directions

  Suppose P and U are points on the line at infinity, given in barycentric coordinates by P =\(p : q : r\) and U =\(u : v : w\). We call P and U perpendicular directions if for every point X not in the line at infinity, the lines XP and XU are perpendicular. (It is well known that if P and U are an antipodal pair on the circumcircle, then their isogonal conjugates are perpendicular directions.)

• X₃₃₁₈–X₃₃₂₈ — Incircle transforms

  Suppose that U =\(u : v : w\) (trilinears) is a point other than the symmedian point, X₆. The incircle transform of :g:`U <U*>` is the point T(U) = a (a - b + c) (a + b - c) (c v - b w)^2 : : , which lies on the incircle. Geometrically, T(U) is the Brisse transform of the psi transform of X₆ and U. (Brisse transform is described just before X₁₂₅₄, and psi transform, also called the “Psi mapping” elsewhere in ETC, and TCCT, p. 80, is defined in the preamble just before X₉₈.

• X₃₄₁₅–X₃₄₄₆ — Vertex conjugates

  Suppose that U =\(u : v : w\) and X =\(x : y : z\) (trilinears) are points not on a sideline of ABC. Let \(f(a,b,c) = a/[a^{2}vwyz - ux(bw + cv)(bz + cy)]\). The U-vertex conjugate of X is the point \(f(a,b,c) : f(b,c,a) : f(c,a,b)\). For a geometric interpretation, let T be the vertex triangle of the circumcevian triangles, A_UB_UC_U and A_XB_XC_X, of U and X; viz., the sidelines of T are A_UA_X, B_UB_X, C_UC_X. Then T is perspective to ABC, and the perspector is the U-vertex conjugate of X.

• X₃₆₀₂–X₃₆₀₉ — Points associated with Morley cubics K29, K30, K31

• X₃₆₁₆–X₃₆₃₇ — Homothetic centers

  Suppose that X is a triangle center, that M is the medial triangle, and that t is a real number. The t-dilation of M from X, denoted by H(X; M, t), is a triangle center. If X =\(x : y : z\) (trilinears), then

  H(X; M, t) = bc[(1 + t)ax + (1 - t)by + (1 - t)cz] : ca[(1 - t)ax + (1 + t)by + (1 - t)cz] : ab[(1 - t)ax + (1 - t)by + (1 + t)cz],

  with inverse given by

  \(H^{-1}(X; M, t)\) = bc[(1 - t)(by + cz) - 2ax] : ca[(1 - t)(cz + ax) - 2by : ab[(1 - t)(ax + by) - 2cz].

  César E. Lozada contributed several such triangle centers (March 21, 2011).

• X₃₆₄₇–X₃₆₅₂ — Kirikami-Schiffler Points

  Suppose that X is a point and A'B'C' is a central triangle. Let L_A be the line through A' parallel to the Euler line of triangle BCX, let L_B be the line through B' parallel to the Euler line of CXA, and let L_C be the line through C' parallel to the Euler line of AXB.

  It is well known that if X=X₁, the incenter, then the three aforementioned Euler lines concur in the Schiffler point, X₂₁. If their parallels, the lines L_A, L_B, L_C concur, the point of concurrence is the Kirikami-Schiffler point of the triangle A'B'C', denoted by KS(A'B'C'). Seiichi Kirikami (February 1, 2011) found that those lines concur if A'B'C' is the reference triangle ABC and also concur if A'B'C' is the medial triangle. Peter Moses found additional cases and properties..

• X₃₆₆₃–X₃₈₆₁ — Central Triangles and More Combos

  Many triangle centers can be defined (as just above) by the form Xcom(nT), where T denotes a central triangle. In order to present such centers, it is helpful to introduce the notation T(f(a,b,c), g(b,c,a)) for central triangles. (This lengthly preamble was completed December 23, 2011.)

• X₃₇₃₉–X₃₇₅₇ — More Combos. Continuing the discussion of points Xcom(T)

  suppose that T is an arbitrary triangle, and let nT denote the normalization of T. Let NT denote the set of these triangles, as matrices, and let * denote matrix multiplication. Then NT is closed under *. Also, NT is closed under matrix inversion. Consequently, (NT, *) is a group, comparable to the group of stochastic matrices. Suppose that T₁ and T₂ are triangles. In many cases, the product T₁*T₂ is well-defined (e.g., TCCT, page 175). However, n(T₁*T₂) may not be n(T₁)*n(T₂) if T₁ and T₂ are not normalized. Therefore, it is important, when dealing with products, to include the “n” if it is intended.

• X₃₇₅₈–X₃₈₆₁ — Central Triangles and More Combos

  (See just above, Centers X₃₆₆₃-X₃₈₆₁.

• X₃₈₆₂–X₄₁₆₂ — Inverse Triangles and More Combos

  The Euler triangle and the 2nd, 3rd, 4th, and 5th Euler triangles are discussed in the preamble to X₃₇₅₈, along with eight other central triangles using the notation T(f(a,b,c), g(b,c,a)). Recall that the inverse of a normalized central triangle T, denoted by Inverse(nT) or Inverse(n(T)), is also a normalized central triangle. Barycentrics for the inverse of each of the 13 triangles and properties of these triangles were given (Peter Moses, December 2011).

• X₄₁₈₃–X₄₂₅₀ — Points on the Euler line

• X₄₂₅₁–X₄₂₉₁ — Points on the Brocard Axis

• X₄₂₉₂–X₄₃₅₆ — Points on the Soddy Line

• X₄₃₅₇–X₄₅₄₈ — More Central Triangles and Combos

• X₅₀₀₀–X₅₀₀₁ — Walsmith Points

  Early in 2012, Russell Walsmith introduced this point and conjectured that it lies on the Euler line. Peter Moses found barycentrics for the point and its inverse-in-the-circumcircle, X₅₀₀₁, and proved that the two points lie on the Euler line.

Part 4

• X₅₃₇₆–X₅₃₈₉ — Points on the Hutson-Moses hyperbola

  The Hutson-Moses hyperbola, introduced at X₅₃₇₅, is given by the following barycentric equation:

  a(a - b)(a - c)yz + b(b - c)(b - a)zx + c(c - a)(c - b)xy = 0.

  This hyperbola has perspector X₁₀₀, center X₅₃₇₅, meets the circumcircle in X₈₉₈ and the Steiner circumellipse in X₆₆₆, and is the isogonal conjugate of the line X₂₄₄X₆₆₅. If X =\(x : y : z\) (barycentrics) is a point on the circumcircle, then the point \(H(X) = x/(a(b - c))\): y/(b(c - a)) : z/(c(a - b)) is on the Hutson-Moses hyperbola.

• X₆₀₀₀–X₆₀₃₁ — Isogonal Conjugates With Respect To Various Triangles

• X₆₀₅₆–X₆₀₆₉ — Lozada Perspectors

  Suppose that A'B'C' be the cevian triangle of a point P. Let L be the line through A' tangent to the incircle, and let A* be the touchpoint; define B* and C* cyclically. Then A*B*C* is perspective to ABC for all P. (César Lozada, May 7, 2014, Anopolis #1466)

  If P =\(p : q : r\) (trilinears), then the perspector is P* = a(b + c - a)\(p^{2}\) : b(a + c - b)\(q^{2}\) : c(a + b - c)\(r^{2}\).

• X₆₀₇₀–X₆₀₈₃ — Points Associated with the Steiner Deltoid

  Suppose that ABC is a triangle. Its Steiner deltoid, SD, is the envelope of the Simson lines of ABC. Peter Moses (September 14, 2014) defined a mapping from (O) to SD; the mapping is here denoted by M and named the Moses-Steiner image.

• X₆₁₀₃–X₆₁₁₁ — the Dao-Moses-Telv Circle and Associated Points

  The Dao-Moses-Telv circle is defined as the circle passing through these six points:

  F₁ = X₁₃, the 1st Fermat point

  F₂ = X₁₄, the 2nd Fermat point

  F₃ = inverse of F₁ in the circumcircle

  F₄ = inverse of F₂ in the circumcircle

  F₅ = inverse of F₁ in the nine-point circle

  F₆ = inverse of F₂ in the nine-point circle

  Dao Thanh Oai noted that these points are concyclic, Telv Cohl gave a proof, and Peter Moses discovered properties described in this preamble. The center of the Dao-Moses-Telv circle is X₁₆₃₇. (Francisco Javier García Capitán, November 3, 2014)

  The Dao-Moses-Telv circle is orthogonal to the circumcircle, the nine-point circle, and all the other circles in their coaxal family. The circle passes through X₅₀₀₀ and X₅₀₀₁, these being the Walsmith point and its inverse in the circumcircle. (Peter Moses, November 6, 2014)

• X₆₁₁₂–X₆₁₂₈ — Moses Radical Circle and Associated Points

  Summarizing notes from Peter Moses (November 7, 2014), the circle here named the Moses radical circle, MRC, is defined as the radical circle of the circumcircle, nine-point circle, and Brocard circle. The center of MRC is X₆₄₇, and, like the Dao-Moses-Telv circle (see X₆₁₀₃, MRC passes through X₅₀₀₀ and X₅₀₀₁.

• X₆₁₂₉–X₆₁₃₆ — Centers of Circles Orthogonal to the Coaxal System of the Circumcircle and Nine-Point Circle

  Continuing Peter Moses’s discussion in the preamble to X₆₁₁₂-X)6128), suppose that P =\(p : q : r\) (barycentrics) is a point other than the circumcenter. Let U(P) be the circle that passes through P and its circumcircle-inverse and is orthogonal to the coaxal system of the circumcircle and nine-point circle. Then X₅₀₀₀ and X₅₀₀₁ lie on U

• X₆₁₃₇–X₆₁₄₀ — Centers of Circles Orthogonal to the Coaxal System of the Circumcircle and Brocard Circle

  Continuing Peter Moses’s discussion in the preambles to X₆₁₁₂ and X₆₁₂₉, suppose that P =\(p : q : r\) (barycentrics) is a point other than the circumcenter. Let V(P) be the circle that passes through P and its circumcircle-inverse and is orthogonal to the Schoute coaxal system; that is, the coaxal system of the circumcircle and Brocard circle. Then X₁₅ and X₁₆ lie on V, and the center of V is the point M(V) =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\) given by …

• X₆₃₃₇–X₆₆₆₁ — and X₆₆₀₀-XX(6661), Symbolic Substitution

  contributed by Clark Kimberling, August 8, 2026. The notation “SS(a → \(S_{A})\)” denotes barycentric symbolic substitution, similar to trilinear symbolic substitution defined at X₃₂₂₁. Symbolic substitution carries lines to lines, conics to conics, and cubics to cubics. For example, since X₁, X₂, X₃, X₄, X₆, X₉, X₅₇, X₁₀₇₃, X₁₂₄₉ lie on the Thomson cubic, their images under SS(a →\(S_{A})\) lie on the cubic given by

  \(x(S^{2}_{C}\)\(y^{2}\) - \(S^{2}_{B}\)\(z^{2})\) + \(y(S^{2}_{A}\)\(z^{2}\) - \(S^{2}_{C}\)\(x^{2})\) + \(z(S^{2}_{A}\)\(x^{2}\) - \(S^{2}_{A}\)\(y^{2})\) = 0.

  The subject of symbolic substitution was introduced in C. Kimberling, “Symbolic substitutions in the transfigured plane of a triangle,” Aequationes Mathematicae 73 (2007) 156-171.

• X₆₃₉₅–X₆₅₀₂ — Centers associated with Lucas triangles and Lucas circles

  contributed by Randy Hutson, February 3, 2015

• X₆₅₇₀–X₆₅₇₇ — Circumeigencenters (CET)

  The eigencenter of a triangle is defined in the Glossary and in TCCT, p. 192. In case the triangle is the circumcevian triangle of a point U =\(u : v : w\) (trilinears), the eigencenter is given by [formula]. This point is here named the circum-eigentransform of U, denoted by CET(U). The point CET(U) lies on the circumcircle. Pairs (i,j) such that CET(X_i) = X_j include the following [list].:

• X₆₆₆₆–X₆₇₂₃ — Centroids of Quadrilaterals

  Suppose that X is a point in the plane of a triangle ABC. The centroid, G,′ of {A, B, C,X} is the complement of the complement of X, hence also the midpoint of X and the complement of X, as well as the Kosnita(X,X₂) point. If X =\(x : y : z\) (barycentrics), then G' = 2x + y + z : x +\(2y : z : x\) + y + 2z. Randy Hutson, February 24, 2015, noted that properties described at X₆₆₀₀ and X₆₆₀₁ hold generally, as follows. Let G″ be the complement of X, and let K_aK_bK_c be the anticevian triangle of G″. The reflection of the cevian triangle of X in G' is perspective to ABC, and the perspector is the anticomplement of the centroid of {K_a,K_b,K_c,G″}. This perspector is also the isogonal conjugate of the TCC-perspector of the isogonal conjugate of G″. (The TCC-perspector of a point is defined in the preamble to X₁₆₀₁.)

• X₆₇₂₄–X₆₇₉₆ — Homologous images

  , contributed by Randy Hutson, March 15, 2015. The term “homologous” means “having the same relation or relative position to”. Let T₁ and T₂ be two triangles, and P a point in the plane of triangle T₁. The point P' in the plane of triangle T₂ is a T₁-to-T₂ homologous image of P if P' has the same relation or relative position to T₂ as P has to T₁. If T₁ and T₂ are similar triangles, this meaning is unambiguous, and we call P' the T₁-to-T₂ similarity image of P. If, however, T₁ and T₂ are not similar, the term “homologous images” can apply to several kinds of mappings. We present three such mappings here.>

  Trilinear image:

  Let P be a point with trilinears \(x : y : z\) wrt T₁. Then the T₁-to-T₂ trilinear image of P is the point P' with trilinears \(x : y : z\) wrt T₂. This is a non-affine collineation, preserving collinearities, but not ratios of distances between collinear points. Parallel lines do not remain parallel under this mapping, so that points on the line at infinity map to finite points. Circumconics of T₁ map to circumconics of T₂ and inconics of T₁ map to inconics of T₂.

  Barycentric image:

  Let P be a point with barycentrics \(x : y : z\) wrt T₁. Then the T₁-to-T₂ barycentric image of P is the point P' with barycentrics \(x : y : z\) wrt T₂. This is an affine collineation, preserving collinearities as well as ratios of distances between collinear points. Parallel lines remain parallel under this mapping, so that the line at infinity maps to itself. Conics map to conics of the same type (ellipses, including circles, map to ellipses, parabolas to parabolas, hyperbolas to hyperbolas). Circumconics of T₁ map to circumconics of T₂ and inconics of T₁ map to inconics of T₂.

  Functional image:

  Let the sidelengths of T₁ be denoted \(a_{1}\), \(b_{1}\), \(c_{1}\) and the sidelengths of T₂ be denoted \(a_{2}\), \(b_{2}\), \(c_{2}\). Let P be a point with coordinates \(f(a_{1},b_{1},c_{1}) : g(b_{1},c_{1},a_{1}) : h(c_{1},a_{1},b_{1})\) with respect to T₁, where f, g, h either have the same degree of homogeneity, or else one is the zero function and the other two have the same degree of homogeneity. (If P is a center of T₁, then f = g = h.) Then the T₁-to-T₂-functional image of P is the point P' with coordinates \(f(a_{2},b_{2},c_{2}) : g(b_{2},c_{2},a_{2}) : h(c_{2},a_{2},b_{2})\) with respect to T₂. Coordinates here can be trilinears or barycentrics, as long as the same system is used for T₁ and T₂. P' serves the same ‘function’ wrt T₂ (e.g., centroid, circumcenter, 1st Brocard point, A-vertex of orthic triangle, etc.) as P serves wrt T₁. For example, the orthic-to-excentral functional image of X₅ = X₄₀, since X₅ and X₄₀ are the respective circumcenters of the orthic and excentral triangles. This is a non-affine collineation, preserving collinearities, but not ratios of distances between collinear points (except where these distance ratios define the point, such as midpoints, reflections, etc.). Parallel lines remain parallel under this mapping, so that the line at infinity maps to itself. Circles map to circles. Other conics map to conics, but not necessarily of the same type (e.g., the MacBeath circumconic of T₁ may be an ellipse, but maps to the MacBeath circumconic of T₂, which may be a hyperbola.) Circumconics of T₁ map to circumconics of T₂ and inconics of T₁ map to inconics of T₂.

Part 5

• X₇₀₀₁–X₇₃₇₃ — Centers associated with extra-triangles

  contributed by Richard Hilton, March 19, 2015. Suppose that f(a,b,c) is a polynomial and that a triangle center X has barycentric coordinates \(f(a,b,c) : f(b,c,a) : f(c,a,b)\). Then the A-extraversion of X is obtained by replacing a by -a in all three of its barycentric coordinates. Likewise, the B-extraversion is obtained by the substitution b → -b, and the C-extraversion by c → -c. The three extraversions are the vertices of a central triangle which we shall call the extra-triangle of X. In the notation introduced in the preamble to X₃₇₅₈, the extra-triangle of X is T(f(-a,b,c), f(b,c,-a)). For centers that have no such polynomial representation, the three extraversions are defined “by construction” [details omitted here]. The focus in this section is on centers for which the extra-triangle is perspective with ABC.

• X₇₅₈₄–X₇₅₈₈ — Endo-homothetic centers

  contributed by César Eliud Lozada, April 22-27, 2015. Suppose that U and V are a pair of homothetic triangles. There is a well-known point, X, called the homothetic center of U and V. For example, the homothetic center of the 1st circumperp and 2nd extouch triangles is X = X₇₅₈₀. Now, we can “view” X from ABC as X₇₅₈₀, or we can “view” X from U, in which case X is X₁₉₉₃-of-U, and by homothety, X is also X₁₉₉₃-of-V. In general, if X = X'-of-U (or equivalently, X = X'-of-V), then the point X' (as a function of the reference triangle ABC) is introduced here as the endo-homothetic center of U and V.

• X₇₅₉₇–X₇₆₀₂ — Touchpoints of pairs of circles

  Tran Quang Hung found that the incircle of the hexyl triangle is tangent to the circumcircle of ABC. The point of tangency, or touchpoint, is X₇₅₉₇. Let S be the set of incircles, circumcircles, and nine-point circles of 60 central triangles listed in MathWorld. César Lozada checked pairs of circles in S for tangency and found several new points, X₇₅₉₈-X₇₆₀₂. He notes (May 10, 2015) that these six points together with points already in ETC account for all touchpoints of pairs of circles in S.

• X₇₆₉₃–X₇₇₀₉ — Similar triangles and centers of similitude

  contributed by César Lozada and Randy Hutson, June 8, 2017. If two similar figures lie in the plane but do not have parallel sides (i.e., they are similar but not homothetic), there exists a center of similitude, also called a self-homologous point, which occupies the same homologous position with respect to the two figures (Johnson 1929, p. 16). See Similitude Center at MathWorld. Algebraically, the center of similitude of two similar triangles U and V is the invariant triangle center under the affine transformation that maps U into V. If U and V are homothetic then their center of similitude coincides with their homothetic center.

• X₇₇₁₃ — Hatzipolakis-Lozada Homothetic Centers

  Antreas Hatzipolakis posed the following construction and related questions (June 10, 2015). Let H_aH_bH_c be the orthic triangle of a triangle ABC and H = X₄. Let A_B be the orthogonal projection of H_a on line HH_b, and define B_c and C_a cyclically. Let A_C be the orthogonal projection of H_a on line HH_c, and define B_A and C_B cyclically. Let P be a point (in the usual sense of a function defined on a subset of the set of points in the plane of an abstract triangle with variable side-lengths a,b,c). Let P_A = P-of-H_aA_BA_C, P_B = P-of-H_bB_CB_A, P_C = P-of-H_cC_AC_B. Let P_AB be the reflection of P_A in line HH_b, and define P_BC and P_CA cyclically. Let P_AC be the reflection of P_A in line HH_c, and define P_BC and P_CA cyclically. Let M_A be the midpoint of P_AB and P_AC, and define M_B and M_C cyclically. Then M_AM_BM_C is homothetic to ABC, and the Euler line of M_AM_BM_C is parallel to the Euler line of ABC.

  César Lozada (June 10, 2015) found that if P =\(p : q : r\) (trilinears), then the homothetic center of M_AM_BM_C and ABC is the point given by

  HL(P) = [(\(a^{2}\) + \(b^{2}\) + \(c^{2})\)p + 2abq + 2acr]/((\(a^{2}\) - \(b^{2}\) - \(c^{2})\) : : = (\(S_{\omega }\)p + abq + acr)/\(S_{A}\) : :

  If P is on the infinity line, then HL(P) = P, and if P lies on the Euler line of ABC, then Euler-line-of-M_AM_BM_C = Euler-line-of-ABC.

• X₇₇₂₀–X₇₇₃₃ — Centers related to the orthocentroidal triangle

  César Lozada (June 18, 2015) finds perspectors, orthologic centers, and parallelogic centers associated with the orthocentroidal triangle. This triangle is defined at X₅₄₇₆ as follows. Let A' be the intersection, other than X₄, of the A-altitude and the orthocentroidal circle, and define B' and C' cyclically. The orthocentroidal triangle, A'B'C', is inversely similar to ABC, with center X₆ of similitude.

• X₇₇₄₅–X₇₇₅₄ — Points on the Euler line

  contributed by Peter Moses, July 6, 2015.

• X₇₇₅₅–X₇₉₉₉ — Centers associated with mixtilinear triangles

  César Lozada (July 6, 2015) introduced the 2nd-to-6th mixtilinear triangles and contributes associated triangle centers.

• X₈₀₀₂–X₈₀₂₁ — Centers associated with the Stammler triangle

  César Lozada (July 22, 2015) introduced centers associated with the Stammler triangle. A'B'C', an equilateral triangle whose vertices are the centers of the Stammler circles; see MathWorld: Stammler Circles.

• X₈₀₁₂–X₈₀₄₂ — Danneels points

  Danneels points are introduced at X₃₀₇₈. The definition is restated here for a point U =\(u : v : w\) (barycentrics): D(U) = \(u^{2}(v + w)\) : \(v^{2}(w + u)\) : \(w^{2}(u + v)\). It is proved at X₃₀₇₈ that if U is on the Euler line of a triangle ABC, then D(U) is also on the Euler line.

• X₈₀₇₅–X₈₁₀₄ — Points associated with tangential-midarc triangles

  This section was contributed by César Eliud Lozada, August 18, 2015. Let ABC be a triangle with incenter I. Let D' and D″ be the points in which the line AI meets the incircle, where D' is the closer of the two points to A. Define E' and F' cyclically, and defined E″ and F″ cyclically. Let A'B'C' be the triangle whose sidelines are the tangents to the incircle at D', E', F'. The triangle A'B'C' has been called the tangential-midarc triangle of ABC (e.g., MathWorld), but here it is the 1st tangential-midarc triangle of :g:`ABC << A B C* >>`. Let A″B″C″ be the triangle whose sidelines are the tangents to the incircle at D″, E″, F″. Triangle A″B″C″ is here introduced as the 2nd tangential-midarc triangle of :g:`ABC << A B C* >>`.

• X₈₁₁₇–X₈₁₄₀ — Endo-homothetic centers

  This section was contributed by César Eliud Lozada, September 7, 2015.

• X₈₁₄₁–X₈₁₅₉ — Centers of central circles

  This section was contributed by César Eliud Lozada, September 22, 2015. There are more than sixty central circles included at MathWorld. Triangles centers X₈₁₄₁-X₈₁₅₅ are centers of such circles that are not among the points X_i for i < 8141.

• X₈₁₆₂–X₈₁₇₁ — Similicenters of pairs of circles

  contributed by Peter Moses, October 5, 2015.

• X₈₁₇₆–X₈₁₈₂ — Centers associated with Van Lamoen circles

  contributed by César Eliud Lozada, October 10, 2015. Let M_aM_bM_c be the medial triangle of ABC, and let G = X₂, the centroid of ABC (and of M_aM_bM_c). It is well known that the circumcenters of triangles GBM_c, GCM_a, GAM_b, GCM_b, GAM_c, GBM_a lie on a circle, called the van Lamoen circle of ABC. See van Lamoen Circle at MathWorld. More generally, let U be any triangle perspective to ABC with perspector P such that the six circumcenters all lie on a circle. That circle is introduced here as the U-van Lamoen circle.

• X₈₁₈₅–X₈₂₅₃ — Centers associated with homothetic pairs of triangles

  contributed by César Eliud Lozada, October 13, 2015.

• X₈₂₅₄–X₈₂₆₃ — Midpoints associated with pedal and antipedal triangles

  contributed by Peter Moses, October 17, 2015. In TCCT, pp. 187-188, it is noted that if ABC is a triangle and P is a point not on a sideline of ABC, then the pedal triangle of P is homothetic to the antipedal triangle of the isogonal conjugate, P', of P, and also that the product of the areas of the two triangles is the square of the area of ABC. Moreover, the vertices of the pedal triangles of P and P' are concyclic, and the center of the circle is the midpoint of P and P'. The following table shows examples of P, P', and their midpoint.

• X₈₂₆₈–X₈₂₇₂ — Centers of apedal conics

  contributed by Clark Kimberling and Peter Moses, October 19, 2015. Suppose that ABC is a triangle and P is a point not on a sideline of ABC. Let P' denote the isogonal conjugate of P. The vertices of the antipedal triangles of P and P' lie on a conic, here denoted by apedal(P). The following table shows examples of P, P', and the center of apedal(P'). Conjecture: For every ABC and P, the conic apedal(P) is a hyperbola.

• X₈₂₇₃–X₈₃₅₀ — Perspectors

  contributed by César Eliud Lozada, Oct. 21, 2015. Perspectors are given for pairs of triangles Many of these triangles were introduced in connection with specific triangle centers in ETC.

• X₈₃₇₂–X₈₄₂₄ — Perspectors of pairs of triangles

  contributed by César Eliud Lozada, November 2, 2015. Central triangles A'B'C' and A″B″C″ are perspective if the lines A'A″, B'B″, C'C″ concur. The point of concurrence is their perspector (or center of perspective). If A'B'C' and A″B″C″ are not only perspective, but homothetic, their perspector is their homothetic center.

• X₈₄₃₁–X₈₅₃₃ — Ceva conjugates

  , contributed by Peter Moses, November 4, 2015. These points on the Neuberg cubic, K001 in Bernard Gibert’s Cubics in the Triangle Plane. Each of these centers is the perspector of two triangles from a list of 25 triangles found in Table 19.1.

• X₈₅₃₇–X₈₅₅₀ — Centers associated with Ehrmann circles

  contributed by César Eliud Lozada, November 9, 2015. In Hyacinthos #6098), December 2, 2002, Jean-Pierre Ehrmann defines a circle as follows. Let P be a point in the plane of ABC and not on the lines BC, CA, AB. Let A_B the the point of intersection of the circle {{P,B,C}} and the line AB. Define A_C symmetrically, and define B_C, B_A, C_A, C_B cyclically. These six points of intersection are on a circle if and only if P = X₆. This circle is here named the Ehrmann circle. The circles {{X₆,B,C}}, {{X₆,C,A}}, {{X₆,A,B,}} are the A-Ehrmann circle, B-Ehrmann circle, C-Ehrmann circle, respectively.

• X₈₅₅₁–X₈₅₇₉ — Coefficient Points of Circles

  contributed by César Eliud Lozada, November 15, 2015. Every circle has a trilinear equation of the following form:

  (Lα + Mβ + Nγ)⋅(aα + bβ + cγ) + K⋅(aβγ + bγα + cαβ) = 0 (1)

  and conversely, if K ≠ 0 then the equation (1) represents a circle.

  A circle Λ is a central circle if \(L : M : N\) is a triangle center and K is a homogeneous symmetric function of (a,b,c); see TCCT, pp 219-226 and CircleFunction at MathWorld). In such a case,\(L : M : N\) is called the coefficient point of the circle Λ. (The coefficients L, M, N, K are homogeneous in a,b,c, so that \(L : M : N\) is unambiguously defined. In other words, there is no loss in assuming that K = 1 in equation (1).)

• X₈₅₈₀–X₈₅₈₃ — Perspectors associated with the Atik triangle

  , contributed by Clark Kimberling and Peter Moses, November 22, 2015. Suppose that V is a point outside a circle (U,u). Let (V,v) be the circle with center V that is orthogonal to (U,u), so that \(v^{2}\) = |UV|² - \(u^{2}\) = power of V with respect to (U,u). Let U_V,A be the circle (V,w) obtained from (U,r) = A-excircle and V = incenter, and let L_A be the radical axis of U_V,A and the A-excircle. Define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The triangle A'B'C' is here named the Atik triangle (Atik being the name of a star). The Atik triangle is perspective to ABC at X₃₀₆₂.

• X₈₆₀₁–X₈₆₁₄ — Points associated with orthocevian triangles

  contributed by Clark Kimberling and Peter Moses, November 22, 2015. Let (O,R) be the circumcircle, and let A'B'C' be the cevian triangle of a point X. Let (O_A,\(r_{A})\) be the circle through B' and C' and orthogonal to (O,R). That is, (O_A,\(r_{A})\) is the circle that passes through the points B' and C' and also their inverses in (O,R). Define (O_B,\(r_{B})\) and (O_C,\(r_{C})\) cyclically. The triangle T(X) = O_AO_BO_C is here named the orthocevian triangle of X.

• X₈₆₁₆–X₈₆₆₅ — Gibert circumtangential conjugates

  contributed by Peter Moses, November 26, 2015, based on notes from Bernard Gibert, November 2, 2015, in connection with the cubic K024. The Gibert circumtangential conjugate of a point U =\(u : v : w\) (barycentrics) is defined as the point

  a^2 (a^2 b^2 c^2 u v - b^2 c^4 u v + a^4 c^2 v^2 - a^2 c^4 v^2 + a^2 b^2 c^2 u w - b^4 c^2 u w - 3 a^2 b^2 c^2 v w + a^4 b^2 w^2 - a^2 b^4 w^2) : :

  A related point is the circumtangential-isogonal conjugate of U, defined by

  a^2 (a^2 c^2 u v - c^4 u v + a^2 c^2 v^2 - c^4 v^2 + a^2 b^2 u w - b^4 u w + a^4 v w - a^2 b^2 v w - a^2 c^2 v w - 2 b^2 c^2 v w + a^2 b^2 w^2 - b^4 w^2) : :

  If U is on the circumcircle, then its Gibert circumtangential conjugate, denoted by M₁(U), is on the Lemoine axis, X₁₈₇X₂₃₇, and the circumtangential-isogonal conjugate, denoted by M₂(U), is on the line at infinity, X₃₀X₅₁₁. Thus, if U is on X₁₈₇X₂₃₇, then M₁(U) is on the circumcircle, and if U is on X₃₀X₅₁₁, then M₂(U) is on the circumcircle. For a selection of circumtangential-isogonal conjugates, see X₈₆₆₆-X₈₇₁₄.

• X₈₆₆₆–X₈₇₁₄ — Circumtangential-isogonal conjugates

  contributed by Peter Moses, November 26, 2015, based on notes from Bernard Gibert, November 2, 2015, in connection with the cubic K024. The circumtangential-isogonal conjugate of U, denoted by M₂(U), is defined by

  a^2 (a^2 c^2 u v - c^4 u v + a^2 c^2 v^2 - c^4 v^2 + a^2 b^2 u w - b^4 u w + a^4 v w - a^2 b^2 v w - a^2 c^2 v w - 2 b^2 c^2 v w + a^2 b^2 w^2 - b^4 w^2) : :

  If U is on the circumcircle, then M₂(U), is on the line at infinity, X₃₀X₅₁₁, so that if U is on X₃₀X₅₁₁, then M₂(U) is on the circumcircle. Related conjugates are M₁(U) defined in the preamble to X₈₆₁₆ and the circumnormal-isogonal conjugate M₃(U) defined in the preamble to X₈₇₁₅.

• X₈₇₁₅–X₈₇₂₅ — Circumnormal-isogonal conjugates

  contributed by Peter Moses, November 28, 2015, based on notes from Bernard Gibert, November 28, 2015, as a sequel to the preceding section on circumtangential-isogonal conjugates.

• X₈₇₂₆–X₈₇₃₄ — Perspectors involving the Ascella triangle

  contributed by Clark Kimberling and Peter Moses, December 3, 2015. Let A' = incircle-inverse of A, and define B' and C' cyclically. Let O_A be the circle {{B,C,B',C'}}, and define O_B and O_C cyclically. The circles O_A, O_B, O_C are orthogonal to the incircle. Let V_A be the center of O_A, and define V_B and V_C cyclically. The triangle V_AV_BV_C is here named the Ascella triangle. The Ascella triangle is also the mid-triangle of the excentral triangle and the intouch triangle.

• X₈₇₃₅–X₈₇₅₆ — Points associated with orthoanticevian triangles

  contributed by Clark Kimberling, Peter Moses, and Randy Hutson, November 9, 2015. Let (O,R) be the circumcircle, and let A'B'C' be the anticevian triangle of a point X. Let (O_A,\(r_{A})\) be the circle through B' and C' and orthogonal to (O,R). That is, (O_A,\(r_{A})\) is the circle that passes through the points B' and C' and also their inverses in (O,R). Define (O_B,\(r_{B})\) and (O_C,\(r_{C})\) cyclically. The triangle T(X) = O_AO_BO_C is here named the orthoanticevian triangle of X. If P =\(p : q : r\) (barycentrics) and T(P) is perspective to ABC, then the perspector is barycentric product of P and the orthocenter;

• X₈₇₈₂–X₈₈₅₅ — Centers associated with Ehramann circles

  contributed by César Eliud Lozada, December 11, 2015. Let ABC be a triangle and let P₁ =\(p_{1} : q_{1} : r_{1}\), P₂ =\(p_{2} : q_{2} : r_{2}\), P₃ =\(p_{3} : q_{3} : r_{3}\) (barycentrics) be three non-collinear points, none on a sideline of ABC. The locus of a point M such that the cevian triangle of M and the triangle T = P₁P₂P₃ are perspective, with perspector denoted by Z(M), is given by

  \(F(x,y,z) =\)δ₁₁(\(r_{1}\)y-\(q_{1}\)z)\(x^{2}\)+δ₂₂(\(p_{2}\)z-\(r_{2}\)x)\(y^{2}\)+ δ₃₃(\(q_{3}\)x-\(p_{3}\)y)\(z^{2}\)+(\(p_{2}\)\(q_{3}\)\(r_{1}\)-\(p_{3}\)\(q_{1}\)\(r_{2})\)xyz = 0 (1)

  where δ_ij is the (i, j)-minor of the vertex matrix of the triangle T.

  If T and ABC are perspective, then \(p_{2}\)\(q_{3}\)\(r_{1}\) - \(p_{3}\)\(q_{1}\)\(r_{2}\) = 0 and, if the equation (1) is non-degenerate, it represents a pivotal cubic with pole Ω(T) and pivot P(T) given by:

  Ω(T) = δ₂₂δ₃₃\(p_{2}\)\(p_{3}\) : δ₃₃δ₁₁\(p_{3}\)\(q_{1}\) : δ₁₁δ₂₂\(p_{2}\)\(r_{1}\)

  P(T) = δ₂₂δ₃₃\(p_{3}\)\(r_{2}\) : δ₃₃δ₁₁\(p_{3}\)\(r_{1}\) : δ₁₁δ₂₂\(p_{2}\)\(r_{1}\)

  This section deals with the poles and pivots of these cubics and the perspectors Z(M) for most of the centers M lying on them.

• X₈₈₅₆–X₈₉₅₀ — Loci associated with selected triangles (2)

  , contributed by César Eliud Lozada, December 11, 2015. Let ABC be a triangle and let P₁ =\(p_{1} : q_{1} : r_{1}\), P₂ =\(p_{2} : q_{2} : r_{2}\), P₃ =\(p_{3} : q_{3} : r_{3}\) (barycentrics) be three non-collinear points, none on a sideline of ABC. The locus of a point M such that the anticevian triangle of M and the triangle T = P₁P₂P₃ are perspective, with perspector denoted by Z(M), is given by

  \(F(x,y,z) = (r_{3}\)δ₃₂\(y^{2}\)-δ₂₃\(q_{2}\)\(z^{2})\)x+(δ₃₁\(r_{3}\)\(x^{2}\)-δ₁₃\(p_{1}\)\(z^{2})\)y+(δ₂₁\(q_{2}\)\(x^{2}\)-δ₁₂\(p_{1}\)\(y^{2})\)z+(\(p_{2}\)\(q_{3}\)\(r_{1}\)-\(p_{3}\)\(q_{1}\)\(r_{2})\)xyz = 0 (1)

  where δ_ij is the (i, j)-minor of the vertex matrix of the triangle T.

  If T and ABC are perspective, then \(p_{2}\)\(q_{3}\)\(r_{1}\) - \(p_{3}\)\(q_{1}\)\(r_{2}\) = 0 and, if the equation (1) is non-degenerate, it represents a pivotal cubic with pole Ω(T) and pivot P(T) given by:

  Ω(T) = \(p_{1}\)δ₁₂δ₁₃ : \(q_{2}\)δ₂₁δ₁₃ : \(r_{3}\)δ₃₁δ₁₂

  P(T) = δ₁₃δ₃₂ : -δ₁₃δ₃₁ : δ₃₁δ₁₂

  This section deals with the poles and pivots of these cubics and the perspectors Z(M) for most of the centers M lying on them.

• X₈₉₅₃–X₈₉₉₈ — Centers associated with quadsquare triangles

  contributed by César Eliud Lozada, December 21, 2015. “In every quadrilateral may be inscribed at least one square having a vertex on each of the four sides. If there is more than one such square in a given quadrilateral, there is an infinite number”. Reference: Hebbert, C. M., Annals of Mathematics, Second Series, Vol. 16, No. 1/4 (1914 - 1915), pp. 38-42.

  Let P be a point in the plane Π of ABC and let ϒ_A = A_CA'_BA'_CA_B be a square inscribed in the quadrilateral ABPC, with A_C on line AB, A'_B on BP, A'_C on PC, and A_B on CA. It is easy to prove, using homogeneous coordinates, that ϒ_A is unique if P does not lie on the conic κ_A given by the trilinear equation

  \(Sa^{2}\)\(u^{2}\) + \(bc(a^{2}\) + S)vw + \(ca(b^{2}\) + S)wu + \(ab(c^{2}\) + S)uv = 0

  Note that B and C lie on κ_A. Define κ_B and κ_C cyclically. If

  P ∈ Π - κ_A∪κ_B∪κ_C (1),

  then there exist unique squares ϒ_A, ϒ_B, ϒ_C inscribed in the quadrilaterals ABPC, BCPA, CAPB, respectively. If P =\(u : v : w\) (trilinears) satisfies (1), then the vertices of ϒ_A are as shown here:

  A_B = (au + bv + cw)uS + (abv + \(S_{C}\)u)\(cw : 0 : bvwS\) + (bcv + \(S_{A}\)w)au

  A_C = (au + bv + cw)uS + (acw + \(S_{B}\)u)bv : cvwS + (bcw + \(S_{A}\)v)au : 0

  A'_B = u(auS + \(bS_{B}\)v) : (S + \(S_{A})\)auv + abcuw + (S + \(S_{C})\)cvw : w(auS + \(bS_{B}\)v)

  A'_C =\(u(auS + cS_{C}w) : v(auS + cS_{C}w) : abcuv\) + (S + \(S_{A})\)auw + (S + \(S_{B})\)bvw

  If P lies on the side BC of ABC then ϒ_A is the A-inner-inscribed square of ABC, as defined at Mathworld: Inner Inscribed Squares Triangle. The square ϒ_A is here named here the A-quadsquare triangle of P. The sidelength of ϒ_A is given by

  L_A = S*Sqrt(abc(2\(uvw(aS_{A}\)u+S(bv + cw)) + bc(v^2 w^2 + u^2 w^2 + u^2 v^2)a))/|(au + cw)(au + bv)S + (avw + buw + cuv)abc|

  Let O_A, O_B, O_C be the centers of the squares ϒ_A, ϒ_B, ϒ_C, respectively. The triangle O_AO_BO_C is here named the P-quadsquares triangle. The vertex O_A are given by

  O_A = a(bcvw + 2Su^2) + (S + \(S_{B})\)buv + (S + \(S_{C})\)cuw : abcuw + (S + \(S_{A})\)auv + (S + \(S_{C})\)cvw : abcuv + (S + \(S_{A})\)auw + (S + \(S_{B})\)bvw,

  and cyclically for O_B and O_C.

  The P-quadsquares triangle is perspective to ABC for all P satisyfing (1). The perspector is given by

  Z(P) = u((S + \(S_{B})\)bv + (S+\(S_{C})\)cw) + abcvw : :

  The transformation P → Z(P) maps lines onto conics. For P on the Euler line, the locus of Z(P) is the conic through X₆, X₄₈₅ and the vertices of the three inner-inscribed-squares triangle.

• X₈₉₉₉–X₉₁₁₁ — Crossdifferences involving X₆

  contributed by Peter J. C. Moses, December 28-31, 2015. Suppose that P =\(p : q : r\) and U =\(u : v : w\) (barycentrics) are points. It is well known that the trilinear pole of the line PU, given by barycentrics 1/(rv - qw) : 1/(pw - ru) : 1/(qu - sv) , lies on the circumconic with perspector P and on the circumconic with perspector U (e.g., the trilinear pole of X₂U is the point 1/(v - w) : 1/(w - u) : 1/(u - v), on the Steiner circumellpse. If P lies on the circumcircle, then P is the trilinear pole of the line X₆Q, where Q = crossdifference of X₆ and the isogonal conjugate of P; e.g., X₉₉ = trilinear pole of X₆X₅₂₄, where X₅₂₄ = crossdifference of X₆ and X₅₁₂.

• X₉₁₁₄–X₉₁₁₇ — Orthologic centers

  contributed by César Eliud Lozada, January 5, 2016. Let L_i and G_c be the triangles T' and T″, respectively, defined at X₉₁₁₂, i.e., L_i is the equilateral triangle with least area inscribed in ABC and G_c is the equilateral triangle with greatest area circumscribed to ABC. Let G_i and L_c be the triangles T' and T″, respectively, defined at X₉₁₁₃, i.e., G_i is the equilateral triangle with greatest area inscribed in ABC and L_c is the equilateral triangle with least area circumscribed to ABC.

• X₉₁₁₈–X₉₁₂₂ — Loud centers

  contributed by César Eliud Lozada, January 8, 2016. In 1891, Frank Herbert Loud published the following theorem:

  Let P₁, P₂, P₃, Q₁, Q₂, Q₃, \(R_{1}\), \(R_{2}\), \(R_{3}\) be nine distinct points on a cubic Ψ such that:

  (1) P₁, P₂, P₃ are collinear on a line, L_P;

  (2) P₁, Q₁, \(R_{1}\) are collinear on a line, L₁;

  (3) P₂, Q₂, \(R_{2}\) are collinear on a line, L₂;

  (4) P₃, Q₃, \(R_{3}\) are collinear on a line, L₃.

  Define \(S_{1}\), \(S_{2}\), \(S_{3}\) and T₁, T₂, T₃ as follows:

  (i) \(S_{1}\) = Q₂\(R_{3}\)∩Q₁\(R_{1}\) and T₁ = Q₂\(R_{3}\)∩Ψ - {Q₂, \(R_{3}\)}

  (ii) \(S_{2}\) = Q₃\(R_{1}\)∩Q₂\(R_{2}\) and T₂ = Q₃\(R_{1}\)∩Ψ - {Q₃, \(R_{1}\)}

  (iii) \(S_{3}\) = Q₁\(R_{2}\)∩Q₃\(R_{3}\) and T₃ = Q₁\(R_{2}\)∩Ψ - {Q₁, \(R_{2}\)}

  Then (a) the points \(S_{1}\), \(S_{2}\) and \(S_{3}\) are collinear (on a line L_S ); (b) the points T₁, T₂ and T₃ are collinear (on a line L_T ); and (c) the lines L_P, L_S and L_T concur.

  Reference: Loud F.H. “A theorem in plane cubics”, Annals of Mathematics, Vol. 6, No. 1 (Jun., 1891), pp. 5-6

• X₉₁₂₃–X₉₂₁₄ — Parry triangles and associated centers

  contributed by Randy Hutson, January 15, 2016. Let A₁ be the intersection, other than X₁₁₁, of the Parry circle and the line AX₁₁₁, and define B₁ and C₁ are cyclically. The triangle A₁B₁C₁, here named the 1st Parry triangle, is similar to ABC and orthogonal to ABC, with similitude center X₁₁₀. Let A₂ be the intersection, other than X₁₁₀, of the Parry circle and line AX₁₁₀, and define B₂ and C₂ cyclically. The triangle A₂B₂C₂, here named the 2nd Parry triangle, is similar to ABC and orthogonal to ABC, with similitude center X₁₁₁. On the Parry circle, A₁B₁C₁ and A₂B₂C₂ are antipodes. The Euler line of A₁B₁C₁ is the Euler line of A₂B₂C₂, and the Brocard axis of A₁B₁C₁ is the Brocard axis of A₂B₂C₂. The Brocard axis of A₁B₁C₁, and also of A₂B₂C₂, is the Lemoine axis of ABC.

  Let A₃ be the intersection, other than X₂, of the Parry circle and the A-median, and define B₃ and C₃ cyclically. The triangle A₃B₃C₃, here named the 3rd Parry triangle, is similar to the 4th Brocard triangle, with similitude center the intersection, other than X₂, of the Parry circle and the orthocentroidal circle. Also, A₃B₃C₃ is the reflection of the circumsymmedial triangle of A₂B₂C₂ in the common Brocard axis of A₁B₁C₁ (which is also the Brocard axis of A₂B₂C₂ and A₃B₃C₃). The triangles A₂B₂C₂ and A₃B₃C₃ are perspective at X₆₄₇. Let A'B'C' be the reflection of the circumsymmedial triangle in the Brocard axis. Then A₃B₃C₃ is similar to A'B'C', with similitude center X₁₁₁.

• X₉₂₁₇–X₉₄₆₁ — Centers related to bicentric pairs

  contributed by César Eliud Lozada - January 17, 2016 and January 19, 2016

• X₉₅₁₇–X₉₅₈₂ — Midpoints on the infinity line

  contributed by Peter Moses, February 11, 2016. Suppose that P =\(p : q : r\) and U =\(u : v : w\) (barycentrics) are distinct points on the line L at infinity; that is, p + q + r = 0 and u + v + w = 0. As in the definition of orthopoint (copied below from Glossary), P may be regarded as a “direction” in the plane of the reference triangle ABC, and all the (parallel) lines in this direction meet in P, and likewise for U. Let L(P) be any line that meets L in P, and let L(U) be any line that meets L in U. Let W₁ and W₂ be the angle bisectors of the angles between L(P) and L(U) at L(P)∩L(U). The midpoints of P and U are here defined as W₁∩L and W₂∩L. Since L(P) and L(U) are perpendicular lines, the two midpoints are a pair of orthopoints, defined in the Glossary of ETC.

• X₉₅₃₄–X₉₇₁₅ — Centers of similitude

  contributed by César Eliud Lozada, February 27, 2016. The appearance of (Φ₂, i, j) in the following lists means that the insimilcenter and the exsimilcenter of the circles Φ₁ and Φ₂ are X_i and X_j, respectively. (Followed by lists.)

• X₉₇₁₈–X₉₇₂₁ — Centers related to Dou circle

  contributed by César Eliud Lozada, March 10, 2016. The Dou circle is defined in MathWorld as the circle cutting the sidelines of the reference triangle ABC at A', A″, B', B″, C', C″ so that angle(A'AA″) = angle(B'BB″) = angle(C'CC″) = 90º. The center of the Dou circle is X₁₅₅.

  Let \(f(a,b,c) = (a^{2})[S_{A} (2 R^{2}-S_{A})\) and \(g(a,b,c) = bc(S^{2}+S_{B} S_{C}-2 (S_{B}+S_{C}) R^{2})\). A trilinear equation for the Dou circle is then

  f(a,b,c)x^2 + f(b,c,a)y^2 + f(c,a,b)z^2 + g(a,b,c)yz + g(b,c,a)zx + g(c,a,b)xy = 0,

  and the radius-squared is (4*R^6-S^2*(5*R^2-SW))/(4*R^2-SW)^2.

  Let S = {circumcircle, nine-points-circle, orthocentroidal circle, orthoptic circle of the Steiner inellipse, polar circle (for obtuse ABC), tangential circle, Taylor circle}, and suppose that U and V are in S. The radical circle of {U, V, Dou circle} is here named the Dow radical circle of U and V, denoted by Dou(U,V). The center of Dou(U,V) is X₂₅₀₁, and the radius-squared of Dou(U,V) is the power of X₂₅₀₁ with repect to U and V. Moreover, Dou(U,V) is orthogonal to the Dou circle and every circle in S. The Stevanovic circle is orthogonal to the Apollonius circle, Bevan circle, excircles-radical circle, as well as every circle in S except the Taylor circle. The radical axis of the Stevanovic and each Dou-radical circle is the Euler line, and their radical trace is X₄₆₈. If ABC is acute then Dou(U,V) passes through X₅₀₀₀ and X₅₀₀₁.

• X₉₇₄₂–X₉₇₇₅ — Artzt triangle

  contributed by César Eliud Lozada, March 25, 2016. The A-Artzt parabola of a triangle ABC is the parabola tangent at B and C to the sidelines AB and AC, respectively. See William Gallatly, The Modern Geometry of the Triangle, 2nd ed. (London: Hodgson, 1913), p. 42.

  The A-Artzt parabola has trilinear equation \(a^{2}\)\(x^{2}\) - 4bcyvz = 0. Its axis is the line \(a(b^{2}\) - \(c^{2})\)x + \(b(a^{2}\) - \(b^{2}\) - \(3c^{2})\)y - (\(a^{2}\) - \(3b^{2}\) - \(c^{2})\)cz = 0, and its directrix is the line (\(a^{2}\) - \(b^{2}\) - \(c^{2})\)ax + \(2bc^{2}\)y + \(2b^{2}\)cz = 0.

  The B- and C-Artzt parabolas are defined cyclically. The foci of the three parabolas are the vertices of the 2nd Brocard triangle.

  The trilinear poles of the directrices of the Artzt parabolas lie on the cubics K675 and K676.

  The triangle A'B'C' bounded by the directrices of the Artzt parabolas is here named the Artzt triangle.

• X₉₇₇₆–X₉₈₁₂ — Conway triangle

  contributed by César Eliud Lozada, April 2, 2016. Let ABC be a a triangle with opposite sidelengths a,b,c, respectively. Let A_B be the point P on line AB such that |BP| = b and B lies between A and P. Define B_C and C_A cyclically. Likewise, let A_C be the point Q on line AC such that |CP| = c and C lies between A and P. Define B_A and C_B cyclically. It is known that these six points lie on the Conway circle, with center X₁ and radius-squared \(r^{2}\) + \(s^{2}\), where r = inradius and s = semiperimeter of ABC.

  Let A' = B_AB_C∩C_AC_B, and define B' and C' cyclically. The triangle A'B'C' is here named the 2nd Conway triangle of ABC. Let A″B″C″ be the (1st) Conway triangle, introduced at X₇₄₁₁, and let A*B*C* be the intouch triangle; A' = reflection of A″ in A*, and likewise for B' and C', as noted by Peter Moses, April 2, 2016.

  area(A'B'C') = 8Rs

  |B'C'| = sqrt(8*S*R*a/((s-b)*(s-c)))

  A' = - (a + b + c) : a + b - c : a - b + c (barycentric coordinates)

  The vertices A', B', C' lie on these cubics: K007 (Lucas cubic), K028, K461, K651.

  Another construction of A_B and A_C follows. Let \(p_{a}\) be the parabola tangent to BC at the A-cevian-trace-of-X₇₅ and also tangent to the sidelines AB and AC. A_B and A_C are the touchpoints of \(p_{a}\) with AB and AC.

  A'B'C' is perspective to ABC with perspector X₇; it is also perspective with perspector X₈ to these triangle: anticomplementary, Fuhrmann and outer-Garcia.

  A'B'C' and Fuhrmann triangles are inversely similar, with X₉₇₈₂ as center of inverse similitude.

• X₉₈₁₃–X₉₈₃₇ — Submedial triangle and related centers

  contributed by César Eliud Lozada, April 15, 2016. Let ABC be a triangle, and suppose that C' is a point on side AB and that B' is a point on side AC. Let r(B'C') be the rectangle whose vertices are B', C', and the orthogonal projections of B' and C' onto side BC. Let \(R_{A}\) be the rectangle r(B'C') of maximal area, which is obtained by taking A'B'C' to be the medial triangle of ABC. Let O_A be the center \(R_{A}\), and define O_B and O_C cyclically. The central triangle O_AO_BO_C is here named the submedial triangle; c.f. the Calabi triangle (https://en.wikipedia.org/wiki/Calabi_triangle).

  The A-vertex of O_AO_BO_C is given by

  O_A = 2*a*b*c : (3*a^2+b^2-c^2)*c : (3*a^2-b^2+c^2)*b (trilinears)

  The vertices of the submedial triangle lie on the cubic K281, and

  X₆₆₈₈ = X₂-of-O_AO_BO_C

  X₃₆₂₈ = X₃-of-O_AO_BO_C

  X₅₄₆₂ = X₄-of-O_AO_BO_C

  The circumcircle of O_AO_BO_C passes through these points: X₆₆₆₇, X₆₇₂₁, X₆₇₂₂, X₆₇₂₃, and the X₅ is the radical center of the circumcircles of the rectangles \(R_{A}\), \(R_{B}\), \(R_{C}\).

• X₉₈₃₃–X₁₀₀₀₀ — Orthologic centers

  contributed by César Eliud Lozada - April 28, 2016.

Part 6

• X₁₀₀₀₁–X₁₀₀₁₄ — Parallels-Conics and related points

  contributed by Peter Moses, May 2, 2016. Let A' be the line through a point P =\(p : q : r\) (barycentrics) parallel to line BC. Let A_B = A'∩AB and A_C = A'∩AC. Define B_C and C_A cyclically, and define B_A and C_B cyclically. The six points A_B, B_C, C_A, A_B, B_C, C_A lie on a conic, here called the parallels-conic of P, denoted by Cpar(P). The point P is here called the base-point of Cpar(P). The center of Cpar(P) is the point W(P) given by

  W(P) = p*(\(p^{2}\) - pq - pr - 2qr) : q*(\(q^{2}\) - qr - qp - 2rp) : r*(\(r^{2}\) - rp - rq - 2pq)

  The perspector of Cpar(P) is the point

  W*(P) = p(2qr + 2pq + pr)(2qr + 2pr + pq) : q(2rp + 2qr + qp)(2rp + 2qp + qr) : r(2pq + 2rp + rq)(2pq + 2rq + rp)

  Cpar(P) is an ellipse, parabola, or hyperbola according as P lies inside, on, or outside the Steiner inellipse.

  Let P' be the reflection of P in W(P). Then P' is the base-point of another conic, Cpar(P'), also having center W(P). For example, if P = X₁, then P' = X₉.

  Randy Hutson observes that W(P) is the midpoint of P and P' = X₂-Ceva conjugate of P. (July 20, 2016)

  If P lies on the orthic axis, then Cpar(P) is a rectangular hyperbola, and the locus of W(P) as P traces the orthic axis a circular cubic.

• X₁₀₀₂₅–X₁₀₀₃₀ — H-transforms and K-transforms

  , contributed by Clark Kimberling and Peter Moses, July 13, 2016. Suppose that R =\(r : s : t\) and U =\(u : v : w\) (barycentrics) are points not on a sideline of a triangle ABC. Let

  L_A be the line of the points \(0 : t : r\) and \(- v : w : u\)

  L_B be the line of the points \(s : 0 : r\) and \(v : - w : u\)

  L_C be the line of the points \(s : t : 0\) and \(v : w : - u\)

  L'_A be the line of the points \(0 : r : s\) and \(- w : u : v\)

  L'_B be the line of the points \(t : 0 : s\) and \(w : - u : v\)

  L'_C be the line of the points \(t : r : 0\) and \(w : u : - v\)

  The lines L_A, L_B, L_C concur in a point, P, and the lines L'_A, L'_B, L'_C concur in a point, P'. If R and U are triangle centers, then P and P' are a pair of bicentric points and PP' is a central line f*x + g*y + h*z = 0, so that the point H(R,U) =\(f : g : h\) is a triangle center, here introduced as the H-transform of R and U, given by first barycentric

  f = v*w*(u/r + v/s - w/t)(u/r - v/s + w/t) - u^2 (v/s + w/t - u/r)^2 .

  Next, let

  L_A be the line of the points \(0 : t : r\) and \(- w : u : v\)

  L_B be the line of the points \(s : 0 : r\) and \(w : - u : v\)

  L_C be the line of the points \(s : t : 0\) and \(w : u : - v\)

  L'_A be the line of the points \(0 : r : s\) and \(- v : w : u\)

  L'_B be the line of the points \(t : 0 : s\) and \(v : - w : u\)

  L'_C be the line of the points \(t : r : 0\) and \(v : w : - u\)

  The lines L_A, L_B, L_C concur in a point, P, and the lines L'_A, L'_B, L'_C concur in a point, P'. If R and U are triangle centers, then P and P' are a pair of bicentric points and PP' is a central line f*x + g*y + h*z = 0, so that the point K(R,U) =\(f : g : h\) is a triangle center, here introduced as the K-transform of R and U, given by first barycentric

  f = v*w*(u/t + w/s - v/r)(u/s + v/t - w/r) - u^2 (v/r + w/s - u/t)(v/t + w/r - u/s)

  Let G = centroid = X₂. Then \(H(G,P) = K(G\),P) and \(H(P,G) = K(P\),G) for all P. For fixed X, the locus of a point P satisfying \(H(P,X) = G is the circumconic with center X\). In particular, for X = X₁₂₅, the locus is the Jerabek hyperbola; for X = X₁₁₅, the Kiepert hyperbola; and for X = X₁₁, the Feuerbach hyperbola.

  If P =\(p : q : r\), then \(H(P,P) = qr - p^{2}\): rp - \(q^{2}\) : pq - \(r^{2}\), which is the Steiner-circumellipse-inverse of P.

  If P is on the circumcircle, then K(P,X₆) = X₃₈₄.

• X₁₀₀₃₇–X₁₀₀₉₄ — Inner- and outer- Yff triangles

  contributed by César Eliud Lozada, August 4, 2016. The Yff circles are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle (see Mathworld). The circles in each triplet have radius \(r_{1}\)=r*R/(R+r) and \(r_{2}\)=r*R/(R-r), respectively. The centers A₁, A₂ for the A-circles of the first and second triplets have respective trilinear coordinates:

  A₁= -(a^4-2*(b^2+b*c+c^2)*a^2+(b^2-c^2)^2)/(2*a^2*b*c) : 1 : 1

  A₂= +(a^4-2*(b^2-b*c+c^2)*a^2+(b^2-c^2)^2)/(2*a^2*b*c) : 1 : 1

  and cyclically B₁, C₁ and B₂, C₂ for the B- and C- circles.

  The triangles T₁=A₁B₁C₁ and T₂=A₂B₂C₂ are known as the inner- and outer- Yff triangles, respectively.

• X₁₀₀₉₇–X₁₀₁₀₃ — Points associated with Dao circles

  contributed by Peter Moses and Clark Kimberling, August 4, 2016, based findings of Dao Thanh Oai. Suppose that P is a point in the plane of a triangle ABC, but not on a sideline (BC, CA, AB). Let P' be the isogonal conjugate of P. Let (O) be the circumcircle of ABC, and let C(P) be the conic through A, B, C,P,P'. Let D be the point in (O)∩C(P) other than A, B, C; let E be a point on (O), other than A, B, C,D, and let E' the point in DE∩C(P), other than D. The points P,P',E,E' lie on a circle, here named the Dao circle of P, denoted by D(P). The point E' is here named the 1st (P,E)-Dao point. (Based on “A generalization of the Sawayama-Thébault theorem”, Dao Thanh Oai, July 21, 2016; ADGEOM 3383)

  Write P =\(p : q : r\) (barycentrics). Then C(P) is given by

  \(b^{2}\)\(c^{2}\)\(p^{2}\)x(y - z) + \(c^{2}\)\(a^{2}\)\(q^{2}\)y(z - x) + \(a^{2}\)\(b^{2}\)\(r^{2}\)z(x - y) = 0.

  The perspector of C(P) is

  \(a^{2}\)\(p(b^{2}\)\(r^{2}\) - \(c^{2}\)\(q^{2})\) : \(b^{2}\)\(q(c^{2}\)\(p^{2}\) - \(a^{2}\)\(r^{2})\) : \(c^{2}\)\(r(a^{2}\)\(q^{2}\) - \(b^{2}\)\(p^{2})\).

  The circle D(P) meets (O) in another point, the 2nd (P,E)-Dao point, and D(P) meets C(P) in another point, the 3rd (P,E)-Dao point. These points are represented by F and F', respectively, in the following examples.

  Example 1. C(X₃) is the Jerabek conic. Taking E = X₁₁₁ gives E' = X₁₀₀₉₇, F = X₅₅₀₅, F' = X₁₀₀₉₈.

  Example 2. Continuing with C(X₃), take E = X₉₈. Then E' = X₈₇₉, F = X₆₇, F' = X₉₃₅.

  Example 3. Continuing with C(X₃), take E = X₁₀₅. Then E' = X₁₀₀₉₉, F = X₁₀₁₀₀, F' = X₁₀₁₀₁; i.e. the 1st, 2nd, 3rd (X₃,X₁₀₅)-Dao points are E', F, F'.

  Example 4. 1st, 2nd, 3rd (X₂,X₁₁₂)-Dao points are X₂₅, X₁₀₁₀₁, X₁₀₁₀₂.

• X₁₀₁₂₉–X₁₀₁₃₆ — ATFF points of pairs of triangles

  contributed by Peter Moses and Clark Kimberling, August 21, 2016, following the example at X₅₆₄₃ contributed by Angel Montesdeoca. Suppose that A'B'C' and A″B″C″ are distinct triangles in the plane of a triangle ABC. The finite fixed point of the affine transformation that carries A'B'C' onto A″B″C″ is here named accordingly and is denoted by ATFF(A'B'C', A″B″C″).

• X₁₀₁₃₇–X₁₀₁₄₈ — Hex2T circles

  contributed by César Eliud Lozada, August 26, 2016. Suppose that T' and T″ are (central) triangles in the plane of a triangle ABC, and let

  A'B'C' = T'-of ABC

  A_aA_bA_c = T″-of-AB'C'

  B_bB_cB_a = T″-of-A'BC'

  C_cC_aC_b = T″ of A'B'C

  Hex2T(T',T″) = hexagon with vertices A_b, A_c, B_c, B_a, C_a, C_b.

  For many choices of T' and T″, the vertices of Hex2T(T',T″) lie on a conic and for a few of them the conic is a circle.

• X₁₀₁₅₃–X₁₀₂₀₀ — Centroidal conics and related centers

  contributed by César Eliud Lozada, August 28, 2016. Let ABC be the reference triangle and U, V, W three points, at least two of them distinct, and let

  G_a = centroid of AVW, and define G_b and G_c cyclically

  G_u = centroid of UBC, and define G_v and G_w cyclically.

  Then the six centroids lie on a (possibly degenerate) conic, and the triangles G_aG_bG_c and G_uG_vG_w are congruent and homothetic, and their homothetic center is the center of the conic.

  The conic is here named the UVW-centroidal conic and also the centroidal conic of UVW. The triangles G_aG_bG_c and G_uG_vG_w are the 1st and 2nd UVW-centroidal triangles and also the

  1st and 2nd centroidal triangles of UVW.

  The center O(UVW) of the centroidal-conic-of-UVW is a triangle center if U, V, W are either triangle centers or vertices of a central triangle. Such center is the centroid of the six centroids above specified.

• X₁₀₂₃₇–X₁₀₂₅₉ — Eulerologic centers

  contributed by César Eliud Lozada, October 2, 2016. Let T'= A'B'C' and T″ = A″B″C″ be triangles. If the Euler lines of A'B″C″, B'C″A″, C'A″B″ concur, then the triangles T' and T″ are (T′, T″)-eulerologic and the point of concurrence is here named the (T′, T″)-eulerologic center. (Definitions given by Antreas Hatzipolakis in Anopolis 3841).

  Note that the existence of the (T',T″)-eulerologic center does not imply the existence of a (T″,T')-eulerologic center.

  Clearly, if two triangles have the same circumcircle then they are mutually eulerologic. Examples:

  (1) The following triangles are inscribed in the circumcircle of ABC, so that each pair are mutually eulerologic: ABC, circummedial, circumorthic, 1st circumperp, 2nd circumperp, circumsymmedial, 3rd mixtilinear, 4th mixtilinear. The eulerologic center of each pair is X₃.

  (2) The following triangles are inscribed in the nine-point circle of ABC, so that each pair are mutually eulerologic: Euler, 2nd Euler, 3rd Euler, 4th Euler, 5th Euler, Feuerbach, medial, orthic. The eulerologic center of each pair is X₅.

• X₁₀₂₇₆–X₁₀₂₈₁ — Feuerbach quadrangle and related centers

  contributed by César Eliud Lozada, October 17, 2016. Let F_A, F_B, F_C be the A-, B-, C- Feuerbach points of ABC, respectively (i.e., the touchpoints of the nine-points-circle and the excircles). Let F_D=X₁₁ be the Feuerbach point of ABC. The cyclic quadrangle QA_F={F_A,F_B,F_C,F_D} is here named the Feuerbach quadrangle of ABC. The centroid of QA_F is X₁₀₂₇₆. Properties:

  1. A maltitude (“midpoint altitude”) is a perpendicular drawn to a side of a quadrilateral from the midpoint of the opposite side. In a cyclic quadrilateral the four maltitudes concur at the anticenter. The anticenter of QA_F is X₁₀₂₇₇.

  2. In a cyclic quadrangle the centroids of the component triangles are the vertices of another cyclic quadrangle. For QA_F this last quadrangle has centroid coinciding with the centroid of QA_F.

  (3) The diagonal triangle A*B*C* of QA_F has vertices with barycentric coordinates:

  A* = {F_A,F_B}∩{F_C,F_D} = -(SB-SC) : SA-SC : SA-SB

  B* = {F_B,F_C}∩{F_A,F_D} = SB-SC : -(SC-SA) : SB-SA

  C* = {F_C,F_A}∩{F_B,F_D} = SC-SB : SC-SA : -(SA-SB)

  A*, B*, C* lie all on the cubics K237, K238, K239, K672.

  A*B*C* has: area=area(ABC)/2, centroid = X₁₀₂₇₈ , circumcenter = X₁₀₂₇₉, orthocenter = X₅ and nine-point-center=X₁₀₂₈₀
  In terms of Chris van Tienhoven’s Encyclopedia of Quadri-Figures (EQF), some centers of QA_F are:

  QA-P₁ = Quadrangle centroid = X₁₀₂₇₆

  QA-P₂ = Euler-Poncelot point = X₁₀₂₇₇ = common point of the nine-point-circles of the component triangles

  QA-P₃ = Gergonne-Steiner Point = X₅ = common point of the midray-circles

  The midray circles of the quadrangle {P₁,P₂,P₃,P₄} are the circumcircles of the triangles M_ijM_ikM_il, for all combinations of (i,j,k,l) in {1,2,3,4}, where M_ij = midpoint of {P_i,P_j}.

• X₁₀₂₉₀–X₁₀₆₀₈ — Points associated with mid-triangles and cross-triangles

  contributed by Randy Hutson, October 22, 2016. Let T₁ = A₁B₁C₁ and T₂ = A₂B₂C₂ be central triangles (or a pair of bicentric triangles) in the plane of a triangle ABC.

  Let A' = midpoint of A₁ and A₂, and define B' and C' cyclically. The triangle A'B'C' is here named the mid-triangle of T₁ and T₂, denoted by MT(T₁,T₂).

  Let A″ = B₁C₂∩C₁B₂, and B″ and C″ cyclically. The triangle A″B″C″ is here named the cross-triangle of T₁ and T₂, denoted by XT(T₁,T₂).

  If T₁ and T₂ are homothetic, then both MT(T₁,T₂) and XT(T₁,T₂) are homothetic to T₁ and T₂.

  If T₁ and T₂ are directly similar, then MT(T₁,T₂) is also directly similar to T₁ and T₂, with the same center of similitude.

  If any pair in {T₁, T₂, XT(T₁,T₂)} are perspective, then every pair in the set are perspective.

  If the vertices of T₁ and T₂ lie on a conic, then XT(T₁,T₂) is degenerate (consisting of 3 collinear points). If T₁ and T₂ are also perspective, XT(T₁,T₂) lies on the polar of the perspector wrt the conic. If T₁ and T₂ are the cevian triangles of P and Q, resp., then XT(T₁,T₂) is degenerate and collinear with P and Q.

  If the vertices of T₂ lie on the respective sidelines of T₁ (e.g., A₂ lies on B₁C₁)), then XT(T₁,T₂) = T₁.

  For many choices of triangles T₁, T₂, T₃,

  (perspector of T₁ and XT(T₂,T₃)) = (perspector of T₂ and XT(T₁,T₃)) = (perspector of T₃ and XT(T₁,T₂)).

  If T₁ is the cevian triangle of P and T₂ is the anticevian triangle of Q, then XT(T₁,T₂) is perspective to ABC, and the perspector is collinear with these 3 points: P, P-Ceva conjugate of Q, Q-cross conjugate of P. Also, XT(T₁,T₂) is perspective to T₁ at Q.

  If T₁ is the circumcevian triangle of P, then XT(ABC,T₁) is perspective to the circumcevian triangle of P*, where P* is the circumcircle-inverse of P. The perspector lies on the circumcircle.

  The cross-triangle of the cevian and anticevian triangles of P is perspective to ABC at P.

  The cross-triangle of the cevian and circumcevian triangles of P is perspective to ABC at gcgP, where g = isogonal conjugate and c = complement.

  The (degenerate) cross-triangle of the circumcevian triangles of P and Q is perspective to ABC at Λ(gP, gQ). Also, the centroid of the (degenerate) cross-triangle of the anticevian triangles of P and Q is the tripolar centroid of the cevapoint of P and Q.

  If T₁ is perspective to ABC at X₂, then the perspector of ABC and XT(ABC,T₁) is the barycentric product A₁*B₁*C₁.

• X₁₀₆₃₁–X₁₀₆₈₂ — Tri-equilateral triangles and related centers

  contributed by César Eliud Lozada, November 5, 2016. As with the Kenmotu squares, we inscribe in a triangle ABC three congruent equilateral triangles PA_bA_c, PB_cB_a and PC_aC_b, with B_a, C_a on BC, C_b, A_b on CA and A_c, B_c on AB. There are two points P making possible this construction: P = P_i=X₁₅ and P = P_o=X₁₆. The equilateral triangles obtained in each case are here named the A-, B-, C- inner/outer equilateral triangles, respectively.

  In each case, the points B_a, C_a, C_b, A_b, A_c, B_c are obviously concyclic. Their circles Γ_i and Γ_o, here named the inner and outer tri-equilateral circles, are denoted and determined as follows:

  Γ_i: center = X₁₅, radius = 2*R/|\(\sqrt{3}\)+cot(ω)|

  Γ_o: center = X₁₆, radius = 2*R/|\(\sqrt{3}\)-cot(ω)|,

  where R and ω are the circumradius and the Brocard angle of ABC, respectively.

• X₁₀₆₉₅–X₁₀₇₀₅ — Reflections of circumcircle-points in the incenter

  contributed by Clark Kimberling and Peter Moses, November 10, 2016. Suppose that P is a point on the circumcircle of a triangle ABC, and let

  P' = reflection of P in the incenter, I, of ABC.

  \(P^{c}\) = complement of P

  \(P^{a}\) = anticomplement of P

  P″ = reflection of X₈ in \(P^{c}\).

  Then

  P' = PI∩\(P^{c}\)X₈

  P' = reflection of X₈ in \(P^{c}\)

  P' = midpoint of X₁₄₅ and \(P^{a}\), where X₁₄₅ = anticomplement of anticomplement of I.

• X₁₀₇₀₆–X₁₀₇₂₀ — Reflections of circumcircle-points in the centroid

  contributed by Clark Kimberling and Peter Moses, November 8, 2016. Suppose that P is a point on the circumcrcle of a triangle ABC, and let

  P' = reflection of P in the centroid, G, of ABC

  \(P^{c}\) = complement of P

  \(P^{a}\) = anticomplement of P.

  Then

  P' = midpoint of G and \(P^{a}\)

  P' = reflection of G in \(P^{c}\)

  P' = {\(P^{c}\), \(P^{a}\)}-harmonic conjugate of P.

• X₁₀₇₂₁–X₁₀₇₂₇ — Reflections of circumcircle-points in the orthocenter

  contributed by Clark Kimberling and Peter Moses, November 9, 2016. Suppose that P is a point on the circumcrcle of a triangle ABC, and let

  H = X₄, the orthocenter of ABC

  P' = reflection of P in H

  \(P^{c}\) = complement of P

  \(P^{a}\) = anticomplement of P

  \(H^{a}\) = anticomplement of H (the orthocenter, X₄ \(H^{aa}\) = anticomplement of \(H^{a}\).

  Then

  P' = midpoint of \(P^{a}\) and \(H^{aa}\)

  P' = reflection of \(H^{a}\) in \(P^{c}\).

• X₁₀₇₃₈–X₁₀₇₅₁ — Reflections of circumcircle-points in the nine-point center

  contributed by Clark Kimberling and Peter Moses, November 10, 2016. Suppose that P is a point on the circumcrcle of a triangle ABC, and let

  O = X₃, the circumcenter of ABC

  N = X₅, the nine-point center of ABC

  P' = reflection of P in N

  \(P^{c}\) = complement of P

  \(P^{a}\) = anticomplement of P.

  Then

  P' = midpoint of H and \(P^{a}\)

  P' = reflection of O in \(P^{c}\).

• X₁₀₇₅₂–X₁₀₇₆₆ — Reflections of circumcircle-points in the symmedian point

  contributed by Clark Kimberling and Peter Moses, November 10, 2016. Suppose that P is a point on the circumcrcle of a triangle ABC, and let

  O = X₃, the circumcenter of ABC

  K = X₆, the symmedian point of ABC

  M = midpoint of \(P^{c}\) and O

  P' = reflection of P in K

  \(P^{c}\) = complement of P

  \(P^{a}\) = anticomplement of P.

  Then

  P' = midpoint of X₁₉₃ and \(P^{a}\)

  P' = reflection of X₆₉ in \(P^{c}\)

  P' = 4M - 3X_10519.

• X₁₀₇₆₇–X₁₀₇₈₂ — Reflections of circumcircle-points in the Feuerbach point

  contributed by Clark Kimberling and Peter Moses, November 10, 2016.

• X₁₀₇₈₃–X₁₀₉₇₆ — Miscellaneous perspectors

  contributed by César Eliud Lozada, November 15, 2016.

• X₁₀₉₇₇–X₁₀₉₈₉ — Centers related to recent advances

  contributed by Randy Hutson, November 15, 2016.

• X₁₁₁₄₇–X₁₁₁₈₈ — Points associated with the anti-Artzt triangle

  contributed by Randy Hutson, December 8, 2016. The anti-Artzt triangle, A'B'C', is here introduced as the triangle of which ABC is the Artzt triangle. A'B'C' is also the anti-McCay triangle of the 1st Brocard triangle. A'B'C' is perspective to ABC at X₅₉₈ and homothetic to the Artzt triangle at X₂. A'B'C' is similar to the circumsymmedial triangle with similitude center X₁₁₀, and inversely similar to the 4th Brocard triangle with center of inverse similitude X₁₁₁₈₇, to the 4th anti-Brocard triangle with center of inverse similitude X₁₉₉₅, and to the McCay and anti-McCay triangles with center of inverse similitude X₂.

  Barycentrics for the A-vertex of the anti-Artzt triangle are given by

  A' = \(b^{2}\) + \(c^{2}\) - \(5a^{2}\) : \(4a^{2}\) + \(4b^{2}\) - \(2c^{2}\) : \(4c^{2}\) + \(4a^{2}\) - \(2b^{2}\)

  A' = SA - 2*SC - 2*SB : SA + SB + 4*SC : SC + SA + 4*SB

• X₁₁₁₈₉–X₁₁₂₆₈ — Centroids and circumcenters associated with central triangles

  Suppose that G = centroid, and that T is a central triangle. Peter Moses has observed that the center of the centroidal conic of T (defined in the preamble to X₁₀₁₅₃ is the midpoint of the segment from G(ABC) to G(T). Equivalently, G(T) is the reflection of G in the center of the conic.

• X₁₁₃₆₃–X₁₁₅₃₆ — Anti-triangles and related centers

  contributed by César Eliud Lozada, December 24, 2016. Several anti-triangles have been previously defined in ETC, including anti-Brocard triangles, anti-McCay and anti-Artzt. Associated with anti-triangles are composite triangles. For example, there are pairs of much-studied triangles whose composite is simply the reference triangle; e.g., the (2nd circumperp triangle of circumorthic triangle) = ABC; and conversely, (circumorthic triangle of 2nd circumperp) = ABC. Further examples of such pairs include these:

  2nd circumperp, circumorthic

  2nd Euler, hexyl

  excentral, orthic

  Johnson, Johnson

  intouch, tangential

• X₁₁₆₁₅–X₁₁₆₃₃ — Circles through X₁₁₁ and related centers

  contributed by César Eliud Lozada, January 8, 2017. Sava Grozdev and Deko Dekov found, by using a computer program, 23 circles (21 of them distinct, 18 of them unnamed until now) all passing through the Parry center X₁₁₁ (See this reference). These circles were verified algebraically and, in the following table, they are described together with their centers. (Followed by a table.)

• X₁₁₆₃₄–X₁₁₆₄₄ — Clifford(4) centers

  contributed by César Eliud Lozada, January 9, 2017. Clifford’s theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles in general position.

  1st theorem: Given four circles \(d_{i}\), i=1..4, passing through a common point M, let P_ij be the second intersection of \(d_{i}\) and \(d_{j}\). Define the circles d’_l={P_ij, P_jk, P_ki} for l=1..4. Then these last four circles have a common point Q, here denoted as the Clifford(4) center of circles \(d_{i}\).

  2nd theorem: Given five circles passing through a common point M, every subset of four of these circles determines a Clifford(4) center Q_i (by the first theorem). Then these five points Q_i lie on a circle S whose center will be denoted as the Clifford(5) center of the given circles.

  3rd theorem: Given six circles passing through a common point M, every subset of five circles determines a circle \(S_{i}\) (by the 2nd theorem). Then these six circles have a common point, denoted here as the Clifford(6) center of the given circles.

  The sequence of theorems can be continued indefinitely. (References: Mathworld and Wikipedia).

• X₁₁₆₇₇–X₁₂₆₉₁ — Inner-Conway triangle and related centers

  contributed by César Eliud Lozada, January 13, 2017. As a variant of the construction of the Conway circle at MathWorld Conway circle, define A_b and A_c inwards; i.e., A_b is on the ray AC, and A_c on the AB, with |AA_b| = |AA_c| = |BC| = a.

  Construct B_c, B_a, C_a, C_b cyclically. The triangle A'B'C' bounded by the lines A_bA_c, B_cB_a, C_aC_b is here named the inner-Conway triangle of ABC, with A-vertex given by trilinears A' = bc : (c - b)c : (b - c)b. (A'B'C' is also the intouch triangle of the anticomplementary triangle.)

• X₁₁₇₅₂–X₁₁₇₉₁ — Centers associated with the Przybyłowski-Bollin configuration

  contributed by Peter Moses, January 22, 2017. In connection with the Przybyłowski-Bollin configuration described at X₁₁₇₅₃, there are four remarkable triangles. Two of them stem from X₁₅, and the other two from X₁₆. For X₁₅, the two triangles are denoted by AiBiCi, associated with the incenter, and A_aB_bC_c, associated with the A-excenter. In order to write barycentrics for the A-vertex of each triangle, let

  U = Sqrt[2(a^2+b^2+c^2+2 \(\sqrt{3}\) S)] = 2 Sqrt[SW+\(\sqrt{3}\) S]. Then

  Ai = a^2 (\(\sqrt{3}\) (-a^2 + b^2 + c^2) + 2 S) : b (\(\sqrt{3}\) b (a^2 - b^2 + c^2) + 2 S (b + U)) : c (\(\sqrt{3}\) c (a^2 + b^2 - c^2) + 2 S (c + U))

  A_a = a^2 (\(\sqrt{3}\) (-a^2 + b^2 + c^2) + 2 S) : b (\(\sqrt{3}\) b (a^2 - b^2 + c^2) + 2 S (b - U)) : c (\(\sqrt{3}\) c (a^2 + b^2 - c^2) + 2 S (c - U))

  The triangles AiBiCi and A_aB_bC_c are perspective, with perspector X₁₅.

  Let O_a = midpoint of Ai and A_a, and define O_b and O_c cyclically. The points Ai, A_a, B, C lie on a circle with center O_a, and the triangle O_aO_bO_c is perspective to the excentral triangle with X₁₁₇₅₂ as perspector.

• X₁₁₈₂₀–X₁₂₀₀₁ — Miscellaneous perspectors and homothetic centers

  contributed by César Eliud Lozada, February 4, 2017. For definitions of triangles, see the index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html).

Part 7

• X₁₂₀₆₄–X₁₂₀₇₉ — X-parabola and related centers

  contributed by César Eliud Lozada, February 27, 2017. Let A*B*C* be the side triangle of the medial and orthic triangles of ABC, and let A'B'C' be the medial triangle of A*B*C*. Then A, B, C, A', B', C' lie on a parabola here named the X-parabola of ABC. Some properties of this parabola are:

  It has barycentric equation: (b^2-c^2)^2*y*z+(c^2-a^2)^2*z*x+(a^2-b^2)^2*x*y=0 It passes through the vertices of the antipedal triangle of X₄₇₇ and centers X₄₇₆, X₅₂₃, X₆₈₅, X₈₅₀, X₈₉₂, X₂₃₉₅, X₂₅₀₁, X₄₀₂₄, X₄₀₃₆, X₄₅₈₁, X₄₆₀₈, X₅₄₆₆, X₈₅₉₉, X₁₀₄₁₂ and X₁₂₀₇₉. Its directrix and the Euler line of ABC are parallel, therefore its axis and the Euler line of ABC are perpendicular. Its 4th intersection with the circumcircles of ABC and A'B'C' is X₄₇₆ The focus and vertex are X₁₂₀₆₄ and X₁₂₀₆₅, respectively. Its axis is the line {523, 5972}, trilinear polar of X₁₂₀₆₆. Its directrix is the line {30, 10279}, trilinear polar of X₁₂₀₆₇. The perspector is X₁₁₅ and the center is X₅₂₃. The dual conic of the X-parabola has center X₆₂₀, perspector X₄₅₉₀ and passes through the vertices of the cevian triangle of X₄₅₉₀ and centers X₂, X₃₂, X₄₃₉, X₅₉₃, X₁₅₀₉, X₂₄₈₂, X₃₉₂₆, X₄₀₂₇, X₇₀₅₈, X₇₇₉₄, X₁₁₁₂₈, X₁₁₁₂₉.

  Let \(t_{a}\), \(t_{b}\), \(t_{c}\) be the tangents to the X-parabola at A, B, C, respectively; the triangle A_tB_tC_t bounded by these tangents is here named the X-parabola-tangential triangle of ABC.

• X₁₂₁₁₀–X₁₂₂₆₉ — Orthologic centers

  contributed by César Eliud Lozada, March 10, 2017.

• X₁₂₂₇₀–X₁₂₄₃₁ — Orthologic centers

  contributed by César Eliud Lozada, March 16, 2017.

• X₁₂₄₃₄–X₁₂₆₂₄ — Orthologic centers

  contributed by César Eliud Lozada, March, 22, 2017.

• X₁₂₆₂₅–X₁₂₈₆₈ — Orthologic centers

  contributed by César Eliud Lozada, March, 26, 2017.

• X₁₂₈₃₅–X₁₂₈₄₁ — Centers associated with the ellipse IE59

  contributed by Peter Moses, March 29, 2017. Let IE59 denote the inellipse with perspector X₅₉. The center of IE59 is X₁₃₀₀₆, and IE59 passes through X_i for these i:

  55, 56, 181, 202, 203, 215, 1124, 1335, 1362, 1397, 1672, 1673, 1682, 2007, 2008, 3235, 3236, 3237, 3238, 6056, 7005, 7006, 7066, 10799, 12835, 12836, 12837, 12838, 12839, 12840, 12841

  This ellipse IE59 is the locus of the centers of similtude (insimilicenter and exsimilicenter) of the incircle with Tucker circles. Also, IE59 intersects the incircle in X₁₃₆₂ and three other points, so that the corresponding four Tucker circles are tangent to the incircle. The Tucker circle through X₁₃₆₂ has the following parameter:

  arccos[(\(t^{2}\) - \(s^{2})\)/(\(t^{2}\) + \(s^{2})\)], where t = r + 4R.

  The centers of the other three Tucker circles are the extraversions of X₉₇₀, and they lie on the Brocard axis. Not only are these circle internally tangent to the incircle, but they are also externally tangent to the two corresponding excircles. In this section, the names for centers X₁₂₈₃₅ to X₁₂₈₄₁, the notation “Tucker (X,p)-circle” represents the Tucker circle with center X and parameter p.

  Let \(f(a,b,c,x,y,z) = b^{4}c^{4}(a - b - c)^{2}(b - c)^{4}x^{2} - 2a^{4}b^{2}c^{2}(a - b)^{2} (a - b + c)(c - a)^{2}(a + b - c)yz\). The ellipse IE59 is given by the barycentric equation f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + \(f(c,a,b,z,x,y) = 0\).

  Possibly the earliest mention of IE59 occurs in TCCT, page 238, in a list of inscribed ellipses; in that list, this ellipse is denoted by W(X₁₁).

• X₁₂₈₄₂–X₁₃₀₀₅ — Orthologic centers

  contributed by César Eliud Lozada, April 1, 2017.

• X₁₃₀₀₇–X₁₃₁₃₅ — Orthologic centers

  contributed by César Eliud Lozada, April, 5, 2017.

• X₁₃₁₆₅–X₁₃₃₂₀ — Parallelogic centers

  contributed by César Eliud Lozada, April, 9, 2017.

• X₁₃₃₂₃–X₁₃₃₅₇ — Centers on X₃X_6 represented by Tucker parameter

  contributed by Peter Moses, April 14, 2017. A Tucker parameter is a function p = p(a,b,c) symmetric and homogeneous of degree zero in a,b,c. A point P with barycentric coordinates (sin A)[cos(A - arccot(p))] lies on the Brocard axis, X₃X₆ and has combo X₃ + ((cot ω)/p)*X₆.

• X₁₃₄₇₆–X₁₃₄₈₂ — Wolfram´s triangle conics perspectors

  contributed by César Eliud Lozada, June 13, 2017. The appearance of (ℭ, n) in the following list means that the perspector of the conic ℭ is X_n: (Brocard inellipse, 6), (De Longchamps ellipse, 13476), (dual of Yff parabola, 514), (Evans conic, 13477), (excentral-hexyl ellipse, 13478), (Feuerbach hyperbola, 650), (Jerabek hyperbola, 647), (Johnson circumconic, 216), (Kiepert hyperbola, 523), (Kiepert parabola, 99), (Lemoine inellipse, 598), (MacBeath circumconic, 3), (MacBeath inconic, 264), (Mandart inellipse, 8), (orthic inconic, 4), (Stammler hyperbola,*), (Steiner circumellipse, 2), (Steiner inellipse, 2), (Thomson-Gibert-Moses hyperbola, 13480), (Yff hyperbola, 13481), (Yff parabola, 190). (The polar triangle of ABC with respect to the Stammler hyperbola is ABC, i.e., ABC is self-polar with respect to the Stammler hyperbola.) For definitions of these conics, see Wolfram’s Triangle Conics. For Thomson-Gibert-Moses hyperbola, see X₅₆₄₂.

• X₁₃₄₉₂–X₁₃₅₆₀ — Cyclologic centers

  contributed by César Eliud Lozada, June 15, 2017.

• X₁₃₆₃₇–X₁₃₇₂₁ — Tri-squares triangles and related centers

  contributed by César Eliud Lozada, July 1, 2017, with notes by Peter Moses, July 7, 2017. Inscribe three squares into a triangle ABC such that each square has two vertices on two distinct sides of ABC and the other vertices of the three squares coincide at the vertices of another triangle A'B'C'. Let the squares be σ_a=B'C'A_cA_b, σ_b=C'A'B_aB_c and σ_c=A'B'C_bC_a with B_a, C_a on BC, C_b, A_b on CA, A_c, B_c on AB and centers A_o, B_o, C_o, respectively. This construction has four solutions. For each case, the triangle A'B'C' will be named here the tri-squares triangle of ABC and the triangle A_oB_oC_o will be referred here as tri-squares-central triangle of ABC.

• X₁₃₇₅₇–X₁₃₈₅₀ — Centers related to the 2nd tri-squares triangles

  contributed by César Eliud Lozada, July 10, 2017. Tri-squares triangles are defined in the preamble of X₁₃₆₃₇. In this section, A'B'C' is the 2nd tri-squares-triangle and A_oB_oC_o is the 2nd tri-squares-central triangle.

• X₁₃₈₇₃–X₁₃₉₉₃ — 4th tri-squares triangles and related centers

  contributed by César Eliud Lozada, July 15, 2017. Tri-squares triangles are defined in the preamble of X₁₃₆₃₇-X₁₃₇₂₁..

Part 8

• X₁₄₀₃₀–X₁₄₀₄₇ — Moses-Euler Points

  On July 21, 2017, Peter Moses noted a relatively simple form for a point P(k) on the Euler line: P(k) = \(b^{4}\) - \(b^{2}\)\(c^{2}\) + \(c^{4}\) + k*(\(a^{4}\) + \(b^{2}\)\(c^{2})\) : : (barycentrics)

  The point is here named the Moses-Euler Point (k = f), where f is a function of (a,b,c), homogeneous of degree 0. The point is given by the combo P(k) = 3(k + 1)(\(a^{4}\) + \(b^{4}\) + \(c^{4})\)*X₂ + 4(k - 1)\(S^{2}\)*X₄.

• X₁₄₁₆₃–X₁₄₁₆₄ — Moses-Yff images

  based on notes from Peter Moses, August 30, 2017. If P =\(p : q : r\) (barycentrics) lies on the circumcircle, then the following point, introduced here as the Moses-Yff image of P, lies on the Yff hyperbola:

  Y(P) = b^2 c^2 (a^4+a^2 b^2-2 b^4-2 a^2 c^2+4 b^2 c^2-2 c^4) (a^4-2 a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) p-a^2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (c^2 (-a^2 b^2+(a^2+b^2-c^2)^2) q+b^2 (-a^2 c^2+(a^2-b^2+c^2)^2) r) : : .

  For example, the Moses-Yff image of X₁₁₀ is X₂, and that of X₇₄ is X₄.

• X₁₄₁₆₉–X₁₄₁₈₈ — Le Viet An equilateral triangles

  contributed by César Eliud Lozada, August 30, 2017. Let ABC be a triangle and BCA', CAB', ABC' equilateral triangles erected out/in - wardly of ABC. Let B_c, C_b be the circumcenters of CC'B and BB'C, respectively, and build C_a, A_c, A_b and B_a cyclically. Denote the circumcenters of B_cC_bA, C_aA_cB, A_bB_aC as O_a, O_b, O_c, respectively. Then, in each case, the triangle O_aO_bO_c is equilateral. (See: Hyacinthos 26551).

  For BCA', CAB', ABC' built outwards ABC, the triangle O_aO_bO_c will be referred here as the outer-Le Viet An triangle.

  This triangle has sidelength O_bO_c = 4*S^2*R*|(SW+\(\sqrt{3}\)*S)/((sqrt (3)*SA+S)*(\(\sqrt{3}\)*SB+S)*(sqrt (3)*SC+S))|.

  O_a has coordinates: O_a = (SW+\(\sqrt{3}\)*S)*a : (SB-SC)*b : (SC-SB)*c (trilinears)
  For BCA', CAB', ABC' built inwards ABC, the triangle O_aO_bO_c will be referred here as the inner-Le Viet An triangle.

  This triangle has sidelength O_bO_c = 4*S^2*R*|(SW-\(\sqrt{3}\)*S)/((sqrt (3)*SA-S)*(\(\sqrt{3}\)*SB-S)*(sqrt (3)*SC-S))|

  O_a has coordinates: O_a = (SW-\(\sqrt{3}\)*S)*a : (SB-SC)*b : (SC-SB)*c (trilinears)

  Both triangles are perspective to the anti-orthocentroidal triangle.

• X₁₄₂₀₆–X₁₄₂₁₃ — Nguyen Images

  Dao Thanh Oai, August 11, 2017. Let ABC be a triangle, let L be the line through B orthogonal to BC, let B_A be any point on L, and let B_C be the point such that \(BB_{A}\)B_CC is a rectangle. Likewise, let \(CC_{B}\)C_AA be a rectangle with base CA and let \(AA_{C}\)A_BB be a rectangle with base AB. Let A' be the midpoint of B_AC_A, and define B' and C' cyclically. Let L_A be the line through A' perpendicular to BC, and define L_B and L_C cyclically. Nguyen Ngoc Giang found that the lines L_A, L_B, L_C concur.

• X₁₄₂₂₆–X₁₄₂₄₅ — Nguyen-Euler centers

  :contributed by César Eliud Lozada, September 4, 2017. This section is based on Nguyen Ngoc Giang’s paper mentioned in the preamble just before X₁₄₂₀₆. The paper appears in the International Journal of Computer Discovered Mathematics, vol 2(2017), pp 135-140.

  Let \(BB_{A}\)C_AC, \(CC_{B}\)A_BA, \(AA_{C}\)B_CB be three rectangles built on the sides of a triangle ABC, such that |BB_A|/a = |CC_B|/b = |AA_C|/c = λ = constant. Assume that λ %gt; 0 means that rectangles are built outwards from ABC and λ < 0 means that rectangles are built inwards from ABC. Nguyen proved that if N_A, N_B, N_C are X₅ (i.e., the nine-point-center) of triangles AB_AC_A, BC_BA_B and CA_CB_C, respectively, then triangles N_AN_BN_C and ABC are orthologic.

  In addition, it can be proved that if the three nine-point-centers (the N’s) are replaced by any other point P on the Euler line such that (X₃P)|/(X₃X₄) = (OP)/(OH) = t, a constant invariant of (a,b,c), then the triangles P_AP_BP_C and ABC remain orthologic. In this case, the orthologic center P_AP_BP_C to ABC is:

  X' = (2*a^6 - (b^2+c^2)*(a^4+(b^2-c^2)^2))*(3*t-1)*λ - 2*S*(a^2+b^2-c^2)*(a^2-b^2+c^2) : : (barycentrics)

  and the orthologic center ABC to P_AP_BP_C:

  X″ = 1/(((2*a^4 + 2*(b^2+c^2)*a^2 + 4*b^2*c^2 - 4*(b^2+c^2)^2)*t + (a^2+b^2+c^2)*(-a^2+b^2+c^2))*λ + 2*S*(-a^2+b^2+c^2)) : : (barycentrics)

  The point X' is here named the Nguyen-Euler(λ) point of P, and the point X″, the Nguyen-Euler(λ) adjoint point of P.

• X₁₄₂₇₂–X₁₄₃₅₃ — Triaxial points

  : contributed by César Eliud Lozada, September 6, 2017. “Let F₁, F₂, F₃ be three figures in perspective two and two in the same plane, show that if they have a common centre of perspective, their three axes of perspective are concurrent.” (Quoted from Lachlan, R.: An Elementary Treatise on Modern Pure Geometry, McMillan & Co., 1893, pp. 123). For three triangles T₁, T₂, T₃ satisfying the those conditions, the point of concurrence of the three axes is here named the triaxial point of the triangles.

• X₁₄₄₅₉–X₁₄₄₇₈ — Koutras-Hatzipolakis-Moses points

  contributed by Peter Moses, September 14, 2017. Following problem 1165a in Stathis Koutras’s posting to Romantics of Geometry, Antreas Hatzipolakis posed the following in Hyacinthos 26601, September 12, 2017:

  Let P be a point in the plane of a triangle ABC. Let

  L_AC = line through A parallel to line CP, and define L_BA and L_CB cyclically

  A' = BC∩L_AC, and define B' and C' cyclically

  M_A = midpoint of A and A', and define M_B and M_C cyclically.

  A* = AM_C∩B'C', and define B* and C* cyclically.

  The points A*, B*, C* are collinear (Koutras).
  L_AB = line through A parallel to line BP, and define L_BC and L_CA cyclically

  A″ = BC∩L_AB, and define B″ and C″ cyclically

  N_A = midpoint of A and A″, and define N_B and N_C cyclically.

  A** = AN_B∩B″C″, and define B** and C** cyclically.

  The points A**, B**, C** are collinear. (Hatzipolakis).

  What can be said about the point of intersection of the lines A*B*C* and A**B**C**? (Hatzipolakis)

  Peter Moses responds as follows. Write P =\(p : q : r\) (barycentrics). The point of intersection, here denoted by KHM(P), lies on the line PX₂ and is given by

  KHM(P) = \(p^{3}\) - \(q^{3}\) - \(r^{3}\) + \(3p^{2}(q + r)\) - 2qr(q + r) - pqr : :

  and by the following combo:

  KHM(P) = 3 (\(p^{3}\) + \(q^{3}\) + \(r^{3}\) + 2 (q r (q + r)+ r p (r + p) + p q (p + q) + 3 p q r))*X₂ - (p + q + r) (2 (\(p^{2}\) + \(q^{2}\) + \(r^{2})\) + 5 (q r + r p + p q))*P

• X₁₄₄₈₃–X₁₄₄₈₈ — Nguyen-Moses images

  contributed by Clark Kimberling and Peter Moses, September 19, 2017. Let ABC be a triangle, let L be the line through B orthogonal to BC, let B_A be any point on L, and let B_C be the point such that \(BB_{A}\)B_CC is a rectangle. Likewise, let \(CC_{B}\)C_AA be a rectangle with base CA and let \(AA_{C}\)A_BB be a rectangle with base AB. Let L_A be the line through A perpendicular to B_CC_B, and define L_B and L_C cyclically. Starting with Nguyen Ngoc Giang’s rectangles (as in the preamble just before X₁₄₂₀₆), Peter Moses found that the lines L_A, L_B, L_C concur.

  Let U be the ratio of the height of the rectangle \(BB_{A}\)B_CC to the base; that is, U = |BA_B|/a. Define V and W cyclically. The point X' of concurrence is given by

  X' = 1 / (-a^2 + b^2 + c^2 + 2 S U) : : (barycentrics)

  If X is a point in the plane of ABC, then it has actual trilinear distances (possibly nonpositive) that are the heights of rectangles as in the above construction. Therefore, starting with X =\(x : y : z\) (barycentrics), we have U = kx/\(a^{2}\), where k = S/(x + y + z), and V = ky/\(b^{2}\) and W = kz/\(c^{2}\). Consequently,

  X' = a^2 / (a^2 (-a^2 + b^2 + c^2) + 2 S^2 x (x + y + z)) : :

  The point X' is here named the Nguyen-Moses image of X.

• X₁₄₅₂₆–X₁₄₅₂₇ — Schiffler triange, Schiffler circles

  contributed by César Lozada, September 23, 2017). Let A'B'C' be the 1st Schiffler triangle of ABC. Let A_a', B_b', C_c' be the orthogonal projections of A, B, C on B'C', C'A', A'B', respectively, and A'_a, B'_b, C'_c the orthogonal projections of A', B', C' on BC, CA, AB. These six points lie on a circle here named the 1st Schiffler circle, with center X₁₄₅₂₆ and squared radius ((9*R+2*r)*R^2*r^2+(R+2*r)*S^2)*R/((R+2*r)^2*(3*R+2*r)^2). This circle passes through X₁₁ and X₁₃₆₅ and is the circumcircle of the pedal triangles of the isogonal conjugate pair X₃₅ and X₇₉.

  Next, if “1st Schiffler triangle” is replaced by “2nd Schiffler triangle”, the resulting six points lie on a circle, here named the 2nd Schiffler circle, with center X₁₇₃₇ and squared radius R*r^2/(R-2*r). This circle passes through X₁₁, X₅₅₃₂, X₁₃₁₄₁, X₁₄₀₂₇ and is the circumcircle of the pedal triangles of the isogonal conjugate pair X₃₆ and X₈₀.

  Continuing, let A'B'C' be the 1st Schiffler triangle and A″B″C″ the 2nd Schiffler triangle of ABC. Let A₁, B₁, C₁ be the orthogonal projections of A', B', C' on B″C″, C″A″, A″B″, respectively, and A₂, B₂, C₂ the orthogonal projections of A″, B″, C″ on B'C', C'A' and A'B'. These six points A₁, B₁, C₁ , A₂, B₂, C₂ lie on a circle here named the 3rd Schiffler circle, with center X₁₄₅₂₇. This circle also passes through X₁₁.

  For definitions of Schiffler triangles, see See César Lozada, Hyacinthos 26620 (IndexOfTrianglesReferencedInETC.html#S)

• X₁₄₅₃₈–X₁₄₅₄₁ — Nguyen orthoimages

  contributed by Peter Moses, September 25, 2017. Let ABC be a triangle, let L be the line through B orthogonal to BC, let B_A be any point on L, and let B_C be the point such that \(BB_{A}\)B_CC is a rectangle. Likewise, let \(CC_{B}\)C_AA be a rectangle with base CA and let \(AA_{C}\)A_BB be a rectangle with base AB. Let A' = A_BC_B∩A_CB_C, and define B' and C' cyclically. Let L_A be the line through A' orthogonal to line BC, and define L_B and L_C cyclically. Nguyen Ngoc Giang found that the lines L_A, L_B, L_C concur. See the preamble just before A(14206).

  Let U be the ratio of the height of the rectangle \(BB_{A}\)B_CC to the base; that is, U = |BA_B|/a. Define V and W cyclically. Peter Moses found (September 23, 2017) that the point X' of concurrence of the lines L_A, L_B, L_C is given by

  X' = \(2a^{2}(a^{2} - b^{2} - c^{2})\)UV + \(2a^{2}(a^{2} - b^{2} - c^{2})\)UW + (\(a^{2}\) + \(b^{2}\) - \(c^{2})\)(\(a^{2}\) - \(b^{2}\) + \(c^{2})\)VW : :

  If X is a point in the plane of ABC, then it has actual trilinear distances (possibly nonpositive) that are the heights of rectangles as in the above construction. Therefore, starting with X =\(x : y : z\) (barycentrics), we have U = kx/\(a^{2}\), where k = S/(x + y + z), and V = ky/\(b^{2}\) and W = kz/\(c^{2}\). Consequently,

  X' = \(2a^{2}\)\(c^{2}(a^{2} - b^{2} - c^{2})\)xy + \(2a^{2}\)\(b^{2}(a^{2} - b^{2} - c^{2})\)xz + \(a^{2}(a^{2} + b^{2} - c^{2})\)(\(a^{2}\) - \(b^{2}\) + \(c^{2})\)yz : :

  X' = \(a^{2}\)(\(S_{B}\)\(S_{C}\)yz - \(b^{2}\)\(S_{A}\)xz - \(c^{2}\)\(S_{A}\)xy : :

  X' = isogonal conjugate of the X₃-vertex conjugate of X

  X' = reflection-in-X₃ of the isogonal conjugate of P

  The triangle A'B'C' is here named the X-Nguyen triangle, and the point X' is named the Nguyen orthoimage of X.

• X₁₄₆₄₉–X₁₄₇₀₆ — Singular Foci of Cubics

  Let O(P) denote the orthopivotal cubic of a point P, as defined at Orthopivotal cubics. Five points on this cubic are A, B, C, X₁₃, and X₁₄. The singular focus of O(P), denoted by Psi(P) defines an involutory mapping P → Psi(P), and Psi(P) is called the Psi-transform of P. The centers X₁₄₆₄₉-X₁₄₇₀₆ involve these orthopivotal cubics for these points: X₁₄₆₅₁, X₁₄₆₅₆, X₁₄₆₆₀, X₁₄₇₀₄, X₁₄₇₀₅, X₁₄₇₀₆. The others involve circular cubics. See Singular Focus of Circular Cubics in Bernard Gibert’s CTC (SingularFocusOfCubics.html).

• X₁₄₇₁₃–X₁₄₇₈₁ — Centers of common circumconics of two triangles

  contributed by César Eliud Lozada, October 10, 2017. The following table shows the ETC indexes of centers of conics circumscribing two or more central triangles. For definitions of triangles see the Index of triangles referenced in ETC (www.faculty.evansville.edu/ck6/encyclopedia/IndexOfTrianglesReferencedInETC.html).

• X₁₄₇₈₂–X₁₄₈₀₆ — Involution and related centers

  contributed by César Eliud Lozada, October 10, 2017. “When several pairs of points A, A'; B, B' ; C, C' ; … lying on a straight line are such that their distances from a fixed point O are connected by the relations

  OA .OA' = OB.OB' = OC.OC' = … ,

  the points are called a range in involution.” (Lachlan, R.: An Elementary Treatise on Modern Pure Geometry. MacMillan & Co., 1893, chapter 5, pp. 37-41).

  A range in involution is denoted by {A, A'}, {B, B'}, {C, C'}. The point O, collinear with the given points, is called the center of involution and each pair of corresponding points, as A and A', are said to be a conjugate-couple.

  To construct O, let {A, A'}, {B, B'} be two pairs of points on a straight line. Through A and B draw any two lines AP, BP intersecting in P and through A', B' draw A'Q, B'Q parallel to BP, AP respectively, meeting in Q. PQ meets AB in O. (Proof in the above reference). From this construction, it is clear that two pairs of points are sufficient for determining the range unambigously and its notation can be shortened to {A, A'}, {B, B'}.

• X₁₄₈₁₅–X₁₄₈₂₅ — Ghoicas-Lozada Images

  contribued by Peter Moses and Clark Kimberling, October 11, 2017. The following material is based on a construction by M. D. Ghoicas in 1934; see Antreas Hatzipolakis and César Lozada, Hyacinthos 26655. Let P be a point in the plane of a triangle ABC, not on one of the three sidelines, and let P* be the isogonal conjugate P*. Let

  A_B = AP∩BC, and define B_C and C_A cyclically.

  A_C = AP*∩BC, and define B_A and C_B cyclically.

  A' = BB_C /CC_B, and define B' and C' cyclically.

  A″ = BB_A /CC_A, and define B″ and C″ cyclically.

  Then the following pairs of triangles are perspective: ABC and A'B'C', ABC and A″B″C″, and A'B'C' and A″B″C″. Let

  Q = ABC and A'B'C'

  Q* = perspector of ABC and A'B'C'

  GL(P) = perspector of A'B'C' and A″B″C″

  The point GL(P) is here named the Ghoicas-Lozada image of P. Let

• X₁₄₈₇₀–X₁₄₈₈₉ — Simpedal points

  contributed by César Eliud Lozada, October 19, 2017. “To determine a point P whose pedal triangle A'B'C' with regard to a given triangle ABC shall be similar to a given triangle A*B*C*”. (Johnson, Roger A.: Advanced Eucliden Geometry, Dover, New York, 1960, problem 205, pp. 142.)

  Assume ABC and A*B*C* are not inversely similar. For finding P, take any two points B₁, C₁ on AC and AB, respectively, and build A₁B₁C₁ directly similar to A*B*C*. Then, through A₂=AA₁∩BC, draw parallels to A₁B₁ and A₁C₁, cutting AC and AB at B₂ and C₂, respectively. The required P is the Miquel point of A₂B₂C₂ with respect to ABC. (Proof in the cited reference).

  The point P and the triangle A'B'C' will be named here the simpedal point of A*B*C* in ABC and the simpedal triangle of A*B*C* in ABC, respectively.

  Let A* = U_A : V_A : W_A (trilinears) and similarly B* and C*. Let M be the trilinear matrix of A*B*C* and \(m_{ij}\) the (i, j)-minor of M. Denote δ_ij = (-1)^i+j \(m_{ij}\). Then,

  P = a (b δ₃₃ δ₂₁ (a^2 - b^2) + c δ₂₂ δ₃₁ (a^2 - c^2) + b c (b δ₃₂ δ₂₁ + c δ₂₃ δ₃₁) - (δ₂₁ δ₃₁ + δ₂₂ δ₃₂ + δ₂₃ δ₃₃) a b c + a δ₂₂ δ₃₃ (b^2 + c^2 - a^2))/(a U_A + b V_A + c W_A) : :

• X₁₄₉₄₁–X₁₄₉₅₂ — Brocard-Lemoine points

  contributed by César Eliud Lozada, October 24-27, 2017. Let ABC be a triangle, P any point in its plane (not on its sidelines) and A', B', C' the traces of P on ABC. Let A_b, A_c be the points where BC is cut by the parallel to AB through B' and the parallel to AC through C', respectively. Build B_c, B_a, C_a, C_b cyclically. Then lines AA_b, BB_c, CC_a concur at a point W₁ and lines AA_c, BB_a, CC_b concur at a point W₂.

  For P =\(u : v : w\) (trilinears), W₁ and W₂ have trilinears as shown:

  W₁ =\(c/v : a/w : b/u\)

  W₂ =\(b/w : c/u : a/v\)

  Note that for P=X₆, W₁ and W₂ are the Brocard points of ABC.

  The previous construction is due to Lemoine as an attempt to generalize the Brocard geometry (Roger A. Johnson, Advanced Eucliden Geometry, Dover, New York, 1960, §499-500, pp. 299). Johnson writes, these points have many properties resembling those of the Brocard points.

  This preamble includes notes from Randy Hutson.

• X₁₄₉₅₃–X₁₄₉₆₆ — Bicentrically induced harmonic conjugates

  contributed by Clark Kimberling and Peter Moses, October 25, 2017. Using barycentric coordinates, suppose that P =\(p : q : r\) and U =\(u : v : w\) are a bicentric pair of points. Let

  P' = line p*x + q*y + r*z = 0

  U' = line u*x + v*y + w*z = 0

  L = a line

  P* = L∩P'

  U* = L∩U'

  X = a point on L

  X' = (P*,U*)-harmonic conjugate of X, to be expressed as L(P,U)-harmonic conjugate of X

  If L is a central line and X is a triangle center, then X' is a triangle center. For example, if X is on the Euler line, then X' is on the Euler line, for every choice of bicentric pair (P,U), and the point X' in this case is written as Euler(P,U)-harmonic conjugate of X. If X is on the Brocard axis, then X' is written as Brocard(P,U)-harmonic conjugate of X.

• X₁₅₀₁₅–X₁₅₁₄₂ — Centers related to altimedial triangles

  contributed by Randy Hutson, October 27, 2017. The term altimedial triangle was introduced in Hyacinthos 253, by Antreas Hatzipolakis, January 30, 2000. A recap follows:

  Let M_A, M_B, M_C be the midpoints of sides BC, CA, AB, resp.

  Let H_A, H_B, H_C be the feet of the altitudes from A, B, C, resp.

  The triangles H_AM_CM_B, M_CH_BM_A, M_BM_AH_C are the A-, B- and C-altimedial triangles. Each is inversely congruent to the medial triangle and inversely similar to ABC (with center of inverse similitude the respective vertex of the orthocentroidal triangle).

  The A-altimedial triangle has barycentric vertex matrix:

  0 : tan B : tan C

  \(1 : 1 : 0\)

  \(1 : 0 : 1\)

  The A-, B- and C-anti-altimedial triangles, A_AB_AC_A, A_BB_BC_B, A_CB_CC_C are here introduced as the triangles of which ABC is the A-, B- and C-altimedial triangle, resp. To construct the A-anti-altimedial triangle, take A_A as the reflection of A in line BC. Then B_A is the reflection of A_A in B, and C_A is the reflection of A_A in C. The B- and C-anti-altimedial triangles are constructed cyclically. The anti-altimedial triangles are each inversely congruent to the anticomplementary triangle and homothetic to the respective altimedial triangle, with center of homothety the respective vertex of the orthocentroidal triangle.

• X₁₅₂₅₄–X₁₅₂₉₉ — Perspeconics

  contributed by César Eliud Lozada, November 14, 2017. Let ABC and A'B'C' be two perspective triangles such that neither is inscribed in the other. Let A_b = BC∩A'B', A_c = BC∩A'C', and likewise for B_c, B_a, C_a, and C_b. As ABC and A'B'C' are perspective, the pairs of lines {BC, B'C'}, {CA, C'A'} and {AB, A'B'} concur at three collinear points, and the lines A_bA_c, B_cB_a, C_aC_b join opposite vertices of an hexagon. By Pascal’s theorem, the six points lie on a conic, here named the perspeconic of ABC and A'B'C'.

• X₁₅₃₄₅–X₁₅₃₄₉ — Dao images

  based on notes from Dao Thanh Oai, November 20, 2017. Let P be an arbitrary point in the plane of a triangle ABC. Let O_A be the centere of the circumcircle of BPC. Let L_A be the line through O_A parallel to line AP, and define L_B and L_C cyclically. The lines L_A, L_B, L_C concur in a point Q = Q(P), here named the Dao image of P. (Dao Thanh Oai, November 20, 2017).

  If P =\(p : q : r\) (barycentrics), then

  Q = p (c^2 q^2 + (-a^2 + b^2 + c^2) q r + b^2 r^2) (2 a^2 q r + p (-(-a^2 + b^2 + c^2) p + (a^2 - b^2 + c^2) q + (a^2 + b^2 - c^2) r)) : :

  If P is on the circumconic Γ = {A, B, C, X₄, P}, then Q is on the complement of Γ. (Peter Moses, November 23, 2017)

  The Dao image of P is the complement of the Kirikami-Euler image of P. (Randy Hutson, December 2, 2017)

  The Dao image of P is the X₂-Ceva conjugate of the 2nd Vu point of P. (Randy Hutson, November 17, 2019)

• X₁₅₄₁₁–X₁₅₄₂₃ — Gibert-Simson Transforms

  This section continues the section on Gibert-Simson transforms, X₂₃₉₄-X₂₄₁₉, with preamble just before X₂₃₉₄. See also Bernard Gibert’s

• X₁₅₄₂₈–X₁₅₄₄₅ — Always-perspective triangles and related centers

  contributed by César Eliud Lozada, December 2, 2017. Centers X₁₆₀₁-X₁₆₃₄ refer to perspectors of the tangential triangle and the circumcevian triangle of an arbitrary point P. This section deals with the perspectivities between a central triangle and a triangle depending on P. Let P =\(u : v : w\) (trilinears) be a point not on the sidelines of ABC and let the reflections-of-P triangle be the triangle whose vertices are the reflections of P in the lines BC, CA, AB. The following pairs of triangle are perspective for all P: ABC-X3-reflections and antipedal-of-P:

  The perspector Q(P) of these triangles is given by trilinears

  Q(P) = b*c*(SA*u^2*(SB^2*v^2+SC^2*w^2)+a*u*v*w*(w*b+v*c)*(S^2-2*SB*SC)-SB*SC*a^2*v^2*w^2)+(S^4-2*SB*SC*(S^2+SA^2))*u^2*v*w : : /

  Q(P) is the reflection in X₃ of the isogonal conjugate of the reflection in X₃ of the isogonal conjugate of P.

  If P is at infinity, Q(P) is the isogonal conjugate of the antipode of the isogonal conjugate of P.

  If P lies on the circumcircle and P' = circumcircle-antipode of P, then Q(SR(X₉₈, P)) = SR(X₉₈, P'), where SR(U,V) is the Simson-Rigby point of U and V. (See the preamble just before X2677 )

• X₁₅₅₃₅–X₁₅₅₅₅ — Lester-Moses Points

  contributed by Peter Moses, December 18, 2017. The Lester circle is the circle that passes through the points X₃, X₅, X₁₃, andf X₁₄. See MathWorld: Lester Circle.

• X₁₅₆₀₇–X₁₅₆₃₉ — Centers related to some problems in chapter 7 of TCCT

  : contributed by César Eliud Lozada, December 22, 2017. This section refers to the general solutions to some selected problems in chapter 7 of TCCT, pp. 211-216.

  Problem 1:

  Let T₁ be the medial triangle and T₂ be the pedal triangle of any point P inside ΔABC. Prove that the side-triangle of T₁ and T₂ is perspective to T₂ and that the center of perspective lies on the nine-points-circle.

  If P =\(u : v : w\) (trilinears) and T₂ is the pedal triangle of P, then the perspector Q(P) is:

  Q(P) = b*c*((b^2-c^2)*u+a*(b*v-c*w))*(c*(-c^2+a^2+b^2)*v-b*(-b^2+c^2+a^2)*w) : : (trilinears)

  The perspector Q(P) is the same point on the nine-points-circle for every point on the line X₃P and it will be denoted here as the (7-1)-TCCT image of the line X₃P .

• X₁₅₆₄₈–X₁₅₆₆₉ — Cross-perspeconics centers

  contributed by César Eliud Lozada, December 28, 2017. Let T₁=A₁B₁C₁ and T₂=A₂B₂C₂ be two perspective triangles, neither inscribed in the other. Let

  1. \(a_{b}\) = A₁B₂, \(a_{c}\) = A₁C₂, \(b_{c}\)= B₁C₂, \(b_{a}\) = B₁A₂, \(c_{a}\) = C₁A₂, \(c_{b}\) = C₁B₂

  2. A_b = \(a_{b}\)∩\(b_{c}\), A_c = \(a_{c}\)∩\(c_{b}\), B_c = \(b_{c}\)∩\(c_{a}\), B_a = \(b_{a}\)∩\(a_{c}\), C_a = \(c_{a}\)∩\(a_{b}\), C_b = \(c_{b}\)∩\(b_{a}\)

  Then the six points A_b, A_c, B_a, B_c, C_a, C_b lie on a conic, here named the cross-perspeconic of T₁ and T₂.

• X₁₅₆₇₀–X₁₅₇₂₃ — Points on the Euler line, by combos

  Certain points on the Euler line are easily represented as combos, as defined in the Introduction, in Part 1. If P and Q are points on the Euler line, then the points h P + k Q, where h and k are constants, comprise the thinline of P and Q, denoted by TL(P,Q). Points X₁₅₆₇₀-X₁₅₆₈₀ are on TL(X₂,X₂₁), and points X₁₅₆₈₁-X₁₅₇₂₃, X₁₅₇₅₉, X₁₅₇₆₄, X₁₅₇₆₅ are on TL(X₂, X₃).

  The Euler line contains infinitely many thinlines, all of which contain the point X₂-X₃ at infinity. Otherwise, distinct thinlines meet in at most one point.

  In general, if P and Q are points, then many combos h P + k Q are composites of the midpoint and reflection operations. Let m = midpoint and r = reflection. Then

  P + Q = m(P,Q)

  P - 2 Q = r(P,Q)

  3 P + Q = m(P,m(P,Q))

  3 P - Q = m(r(Q,P)),P)

  3 P - 2 Q = r(r(P,Q)),P)

  3 P - 4 Q = r(P,r(P,Q))

  4 P - 5 Q = r(r(Q,P),r(P,Q))

  5 P + 3 Q = m(m(m(P,Q),r(Q,P)),Q)

  An open question is whether h*P+k*Q is such a composite for all choices h, k or integers (not both 0). Note that TL(P,Q) also contains points such as P + \(2^{1/2}\)*Q which is not such a composite.

• X₁₅₇₆₆ — Euler lines and Brocard axis concurrences

  :contributed by César Eliud Lozada, January 02, 2018. Let r be the Euler line of ΔABC and P a variable point. Let \(r_{a}\), \(r_{b}\), \(r_{c}\) be the Euler lines of the triangulation of P, here defined as ordered triple (ΔPBC, ΔPCA and ΔPAB) of triangles. Then, if P is on the Neuberg cubic, the four lines r, \(r_{a}\), \(r_{b}\), \(r_{c}\) concur. Under the same conditions, the Brocard axes of ΔABC, ΔPBC, ΔPCA and ΔPAB concur.

• X₁₅₈₀₈–X₁₅₈₈₆ — Tangential perspeconics

  contributed by César Eliud Lozada, January 8, 2018. If A'B'C' and A″B″C″ are two perspective triangles, neither inscribed in the other, then the six lines A'B″, A'C″, B'C″, B'A″, C'A″ and C'B″ are tangent to a unique conic, here named the tangential perspeconic of A’B’C’ and A”B”C”.

  Suppose the first triangle is the reference triangle ABC and the second is a central triangle A'B'C', perspective to ABC at X =\(x : y : z\) (barycentrics). Then XA'/XA = λ_a, XB'/XB = λ_b and XC'/XC = λ_c, where λ_a = λ(a,b,c), λ_b = λ(b,c,a), λ_c = λ(c,a,b) and λ(a,b,c) is a degree-0 homogeneous function of a,b,c. In this case, the center Q of the tangential perspeconic of ABC and A'B'C' is:

  Q = λ_a (λ_b λ_c - 1) (y + z) + (λ_b + λ_c - 2) x : : (barycentrics)

• X₁₅₉₁₅–X₁₅₉₂₈ — Gamma-triangles

  contributed by César Eliud Lozada, January 20, 2018. The gamma triangle Γ(T) of a triangle T is defined as the triangle having vertices the isogonal conjugates of the orthopoints of the sidelines of T. (TCCT, 6-46, pp. 178-179). From this definition, it is clear that every member of a set of homothetic triangles has the same gamma-triangle. It is proved in the given reference that if T is not a degenerated triangle then T and Γ(T) are similar. Their center of similitude will be named here as the gamma-center of similitude of T.

• X₁₅₉₉₅–X₁₅₉₉₉ — Points associated with the Garcia reflection triangle

  This preamble is based on notes from several contributors named in the preamble. The Garcia reflection triangle is introduced here as follows. Let A' be the excenter of a triangle ABC, and define B' and C' cyclically. Let A″ be the midpoint of segment BC, and define B″ and C″ cyclically. Let A* be the reflection of A' in A″, and define B* and C* cyclically. Emmanuel Jose Garcia conjectured that the triangle A*B*C*, here named the Garcia reflection triangle, is perspective to the outer-Garcia triangle. César Lozada proved (January 28, 2018) that the conjecture is true. He found barycentric coordinates:

  A* = a : c - a : b - a

  B* = c \(- b : b : a\) - b

  C* = b - c : a - c : c

Part 9

• X₁₆₁₁₂–X₁₆₁₆₂ — 2nd Fuhrmann triangle and related centers

  contributed by César Eliud Lozada, February 13, 2018. Let ABC be a triangle and let A₁ be the midpoint of the arc BC of its circumcircle not containing A. Define B' and C' cyclically. The triangle A'B'C' is called the Fuhrmann triangle of ABC. (See Fuhrmann triangle from MathWorld). The Furhrman triangle is the reflection of the 2nd circumperp triangle in the sidelines of ABC.

  Let ABC be a triangle and let A″ be the midpoint of the arc BC containing A, and define B″ and C″ cyclically.. The triangle A″B″C″ is here named the 2nd Fuhrmann triangle of ABC. This triangle is the reflection of the 1st circumperp triangle in the sidelines of ABC. Its A-vertex has barycentric coordinates:

  A″ = -a^2 : b*c+a^2-c^2 : b*c+a^2-b^2

  A″B″C″ has area S*(2*r+3*R)/(4*r), where r, R and S are the inradius, circumradius and double-area of ABC, respectively.

  The 2nd Fuhrmann triangle is perspective to triangles in the following list with perspector X₃: anti-Hutson intouch, anti-incircle-circles, 6th anti-mixtilinear, Ara, Ascella, 1st Brocard, 1st circumperp, 2nd circumperp, 1st Ehrmann, 2nd Euler, Fuhrmann, Johnson, inner-Johnson, outer-Johnson, Kosnita, McCay, medial, inner-Napoleon, outer-Napoleon, 1st Neuberg, 2nd Neuberg, tangential, Trinh, inner-Vecten, outer-Vecten. Also, A″B″C″ is perspective to the inner- and outer-Johnson triangles with perspectors X₁₆₁₁₂ and X₁₂₆₃₅, respectively.

• X₁₆₁₈₉–X₁₆₂₂₀ — Eulerologic centers 2

  contributed by César Eliud Lozada, February 19, 2018. Eulerologic triangles and centers are defined in the preamble just before X10237.

• X₁₆₂₈₆–X₁₆₃₀₂ — Collineation images on the Euler line

  A regular collineation m is defined by its action on four points, P₁, P₂, P₃, P₄, no three of which are collinear. Given four such points, let P_k = m(P_k-4) for k = 5,6,7,8. The eight points P_k determine m; conversely, given eight such points, a collineation m is uniquely determined. For every point X, the point m(X) is here called the (P₁, P₂, P₃, P₄; P₅, P₆, P₇, P₈) collineation image of X. (Regular collineations are discussed in Clark Kimberling, Collineations, Conjugacies and Cubics).

  Collineations map lines to lines. Thus, for example the collineation m indicated by (X₁,X₂,X₃,X₆; X₂,X₃,X₆,X₁) maps the Nagel line, X₁X₂, onto the Euler line, X₂X₃. Examples of triangle centers m(X) found in this way are X₁₆₂₈₆-X₁₆₃₀₂. For more collineation images on the Euler line, see X₁₆₃₄₂-X₁₆₃₅₅, X₁₆₃₆₇-X₁₆₃₈₄, and X₁₆₃₉₃-X₁₆₄₅₈. See also the preamble just before X₁₆₅₄₄.

• X₁₆₅₄₄–X₁₆₆₁₄ — Collineation images

  This section follows the discussion of collineations just before X₁₆₂₈₆. If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A, B, C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A, B, C,P). The collineation images at X₁₆₅₄₄-X₁₆₅₇₆ result from A'B'C' = excentral triangle, P = X₂, and U = X₁. We write the image of X as m(X); let \(m^{-1}\) denote the inverse collineation. Then centers X₁₆₅₄₄-X₁₆₅₇₆ are examples of m(X), and X₁₆₅₇₇-X₁₆₆₁₄ are examples of \(m^{-1}(X)\). Other examples are given by the following list …

• X₁₆₆₇₈–X₁₆₇₅₉ — Collineation images

  This section follows the discussion of collineations just before X₁₆₂₈₆, and especially just before X₁₆₅₄₄. If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A, B, C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A, B, C,P).

• X₁₇₀₄₃–X₁₇₀₇₇ — Collineation images

  If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A, B, C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A, B, C,P). The collineation images at X₁₇₀₄₃-X₁₇₀₉₇ result from A'B'C' = medial triangle, P = X₂, and U = X₁. We write the image of X as m(X); let \(m^{-1}\) denote the inverse collineation. Then centers X₁₇₀₄₃-X₁₇₀₇₃ are examples of m(X), and X₁₇₀₇₄-X₁₇₀₉₇ are examples of \(m^{-1}(X)\).

• X₁₇₁₃₄–X₁₇₂₁₉ — Collineation images

  If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A, B, C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A, B, C,P). The collineation images at X₁₇₁₃₄-X₁₇₂₁₉ result from A'B'C' = medial triangle, P = X₂, and U = X₁. We write the image of X as m(X); let \(m^{-1}\) denote the inverse collineation. Then centers X₁₇₁₃₄-X₁₇₁₆₆ are examples of m(X), and X₁₇₁₆₇-X₁₇₂₁₉ are examples of \(m^{-1}(X)\).

• X₁₇₂₂₇–X₁₇₄₀₀ — Moses points

  Consider the set triangle centers having first barycentric of the form

  k(1)*a^2 + k(2)*(ab + ac) + k(3)(b^2 + c^2) + k(4)*bc,

  where k(i) is in {-2,-1,0,1,2} for i = 1,2,3,4. There are 273 such centers, of which 99 are listed above. The remaining 174 appear below as X₁₇₂₂₇ to X₁₇₄₀₀, with names of the form MOSES (k(1),k(2),k(3),k(4)) POINT. Barycentrics contributed by Peter Moses, April 8, 2018.

  We refer to ordered tuples such as (k(1),k(2),k(3),k(4)) as the Moses code for a point X. If X has a first barycentric that is a polynomial in a,b,c, then the Moses notation can be generalized in a straightforward manner, as in X₁₇₆₆₉-X₁₇₆₉₈. The notation can also be extended to cover certain nonpolynomials; e.g. X₆₁₅ can be coded as the Moses (1,0,0,0;-2S) point.

  Specifically, for polynomial centers of degree 3, the Moses code is the 6-tuple (k(1),…,k(6)) of coefficients in

  k(1)*a^3 + k(2)*a^2*(b + c) + k(3)*a*(b^2 + c^2) + k(4)*a*b*c + k(5)*(b^3 + c^3) + k(6)*(b^2*c + b*c^2); see, for example, X₁₇₅₉₁.

  For degree 5, the Moses code is the 12-tuple (k(1),…,k(12)) representing

  k(1)*a^5 + k(2)*a^4*(b + c) + k(3)*a^3*(b^2 + c^2) + k(4)*a^2*b*c + k(5)*a^2*(b^3 + c^3) + k(6)*a^2*(b^2*c + b*c^2) + k(7)*a*(b^4 + c^4) + k(8)*a*(b^3 c + b*c^3) + a(9)*a*b^2*c^2 + a(10)*(b^5 + c^5) + a(11)(b^4 c + b*c^4) + a(12)*(b^3 c^2 + b^2 c^3); see for example, X₁₇₇₈₇.

• X₁₇₄₁₀–X₁₇₄₃₆ — Homo-perspeconics

  contributed by César Eliud Lozada, April 10, 2018. Let :g:`ABC << A B C >>` and :g:`A’B’C’ << A’ B’ C’ >>` be two not homothetic triangles. Let :g:`A″B″C″ << A″ B″ C″ >>` be any other triangle perspective to :g:`ABC << A B C >>` and homothetic to :g:`A’B’C’ << A’ B’ C’ >>`. Then the perspectors of :g:`ABC << A B C >>` and :g:`A″B″C″ << A″ B″ C″ >>` lie on a fixed circumconic of :g:`ABC << A B C >>`. (:person:`Dao Thanh Oai <dao-thanh-oai>`, March 17, 2018). (A proof of this theorem is given in Homo-Perspeconics (Homo-Perspeconics.pdf).)

  This circumconic is here named the homo-perspeconic of A′B′C′ to ABC. It is obvious that if two triangles are homothetic then their homo-perspeconics to ABC are the same.

  The following lists show triangles grouped by their homo-perspeconics to ABC:

  The homo-perspeconic of the following triangles to ABC is the circumcircle: 1st Parry, 2nd Parry

  The homo-perspeconic of the following triangles to ABC is the Feuerbach hyperbola: Ascella, Atik, 1st circumperp, 2nd circumperp, inner-Conway, Conway, 2nd Conway, 3rd Conway, 3rd Euler, 4th Euler, excenters-midpoints, excenters-reflections, excentral, extouch, 2nd extouch, Fuhrmann, 2nd Fuhrmann, inner-Garcia, Garcia-reflection, hexyl, Honsberger, Hutson extouch, inner-Hutson, Hutson intouch, outer-Hutson, incircle-circles, intouch, inverse-in-incircle, 6th mixtilinear, 2nd Pamfilos-Zhou, 1st Schiffler, 2nd Schiffler, 1st Sharygin, tangential-midarc, 2nd tangential-midarc, Yff central

  The homo-perspeconic of the following triangles to ABC is the Kiepert hyperbola: anti-Artzt, 1st anti-Brocard, 6th anti-Brocard, anti-McCay, Artzt, 1st Brocard, 6th Brocard, inner-Fermat, outer-Fermat, 3rd Fermat-Dao, 4th Fermat-Dao, 7th Fermat-Dao, 8th Fermat-Dao, 11th Fermat-Dao, 12th Fermat-Dao, 15th Fermat-Dao, 16th Fermat-Dao, 1st isodynamic-Dao, 2nd isodynamic-Dao, 1st Lemoine-Dao, 2nd Lemoine-Dao, McCay, inner-Napoleon, outer-Napoleon, 1st Neuberg, 2nd Neuberg, 1st tri-squares-central, 2nd tri-squares-central, 1st tri-squares, 2nd tri-squares, 3rd tri-squares, 4th tri-squares, inner-Vecten, outer-Vecten

  The homo-perspeconic of the following triangles to ABC is the Jerabek hyperbola: anti-Ascella, 1st anti-circumperp, anti-Conway, 2nd anti-Conway, 3rd anti-Euler, 4th anti-Euler, anti-excenters-reflections, anti-Hutson intouch, anti-incircle-circles, anti-inverse-in-incircle, 6th anti-mixtilinear, anti-orthocentroidal, Aries, circumorthic, 1st Ehrmann, 2nd Ehrmann, 2nd Euler, extangents, 3rd Hatzipolakis, 1st Hyacinth, 2nd Hyacinth, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, midheight, orthic, orthocentroidal, reflection, submedial, tangential, inner tri-equilateral, outer tri-equilateral, Trinh

  The homo-perspeconic of the following triangles to ABC is the Morley conic: (see X16271) 1st Morley, 2nd Morley, 3rd Morley, Roussel, Stammler

• X₁₇₄₃₈–X₁₇₄₉₉ — Collineation images

  If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A, B, C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A, B, C,P). The collineation images at X₁₇₄₃₈-X₁₇₄₉₉ result from A'B'C' = incentral triangle, P = X₂, and U = X₁. We write the image of X as m(X); let \(m^{-1}\) denote the inverse collineation. Then centers X₁₇₄₃₈-X₁₇₄₇₈ are examples of m(X), and X₁₇₄₇₉-X₁₇₄₉₉ are examples of \(m^{-1}(X)\).

• X₁₇₆₀₃–X₁₇₆₆₈ — Ursa-minor and Ursa-major triangles and related centers

  contributed by César Eliud Lozada, April 14, 2018. Lemma: The internal tangents of two exterior circles touch them in four concyclic points. Also, the external tangents of two circles, none totally interior to the other, touch the circles in four concyclic points.

  In a triangle ABC, the common internal tangents of the incircle and the A-excircle touch them in four concyclic points. Let {\(o_{a}\)} be the circle through these touchpoints and denote as a′ the radical axis of {\(o_{a}\)} and the incircle. Build b′ and c′ cyclically. The triangle A'B'C' bounded by a′, b′ and c′ will be named here the Ursa-minor triangle of ABC.

  Continuing with the previous construction, let a″ be the radical axis of {\(o_{a}\)} and the A-excircle and build b″ and c″ cyclically. The triangle A″B″C″ bounded by a″, b″ and c″ will be named here the Ursa-major triangle of ABC.

  A-vertices of both triangles have trilinear coordinates:

  A' = (b+c)*a-(b-c)^2 : -(a-b+c)*(a-c) : -(a+b-c)*(a-b)

  A″ = (b+c)*(a^2+b^2+c^2)-2*(b^2+c^2)*a : -((a-b)^2+(2*a-c)*c)*(a-c) : -((a-c)^2+(2*a-b)*b)*(a-b)

• X₁₇₆₆₉–X₁₇₆₉₈ — Moses points on the Euler line

  based on notes from Peter Moses, April 14, 2018. Consider a triangle center X having first barycentric of the form

  k(1) a^4 + k(2) a^3 (b + c) + k(3) a^2 (b^2 + c^2) + k(4) a^2 b c + k(5) a (b^3 + c^3) + k(6) a(b^2 c + b c^2) + k(7) (b^4 + c^4) + k(8) (b^3 c + b c^3) + k(9) b^2 c^2,

  where k(i) is in {-3,-2,-1,0,1,2,3} for k = 1, … , 9. Such a point X is here named the MOSES (k(1), k(2), k(3), k(4), k(5), k(6), k(7), k(8), k(9)) POINT. Extending the coding system introduced in the preamble just before X₁₇₂₂₇, we refer to the 9-tuple (k(1),k(2),…,k(9)) as the Moses code for X.

• X₁₇₈₀₇–X₁₇₈₅₀ — Excosine triangles and related centers

  contributed by César Eliud Lozada, April 20, 2018. If the tangents at :g:`B <B>` and :g:`C <C>` to the circumcircle of a triangle :g:`ABC << A B C >>` intersect at :g:`A’ <A’>`, then the circle with center :g:`A’ <A’>` passing through :g:`B <B>` and :g:`C <C>` is called A-excosine circle of :g:`ABC << A B C >>`. This circle cuts :g:`AB <[ A B ]>` and :g:`AC <[ A C ]>` again at two points which are the extremities of a diameter of it.. (Reference: Weisstein, Eric W. “Excosine Circle.” From MathWorld)

Part 10

• X₁₈₂₁₄–X₁₈₂₆₁ — Zaniah triangles and related centers

  contributed by César Eliud Lozada, May 01, 2018. In a triangle ABC, let A_mB_mC_m be the medial triangle, I the incenter and A' the touchpoint of the incircle with the side BC. Then the lines AA', IA_m and B_mC_m are concurrent at a point A₁. Denote B₁ and C₁ cyclically. The triangle A₁B₁C₁ is here named the 1st Zaniah triangle of ABC.

  Continuing with the previous construction, let J_a be the A-excenter and A″ the touchpoint of the A-excircle with the side BC. Then the lines AA″, J_aA_m and B_mC_m are concurrent at a point A₂. Denote B₂ and C₂ cyclically. The triangle A₂B₂C₂ is here named the 2nd Zaniah triangle of ABC.

• X₁₈₃₀₀–X₁₈₅₈₇ — Triangles and centers related to the Ehrmann pivots

  contributed by Randy Hutson, May 2, 2018. Let T_C1=A₁B₁C₁ be the triangle obtained by rotating ABC about the 1st Ehrmann pivot, P(5), by an angle of 2π/3, so that T_C1 circumscribes ABC. Triangle T_C1 is here named the 1st Ehrmann circumscribing triangle. Let T_C2=A₂B₂C₂ be the triangle obtained by rotating ABC about the 2nd Ehrmann pivot, U(5), by an angle of -2π/3, so that T_C2 circumscribes ABC. Triangle T_C2 is here named the 2nd Ehrmann circumscribing triangle. T_C1 and T_C2 are inscribed in the Johnson circle (centered at X₄).

  Let T_I1=A'₁B'₁C'₁ be the triangle obtained by rotating ABC about the 1st Ehrmann pivot, P(5), by an angle of -2π/3, so that T_I1 is inscribed in ABC. Triangle T_I1 is here named the 1st Ehrmann inscribed triangle. Let T_I2=A'₂B'₂C'₂ be the triangle obtained by rotating ABC about the 2nd Ehrmann pivot, U(5), by an angle of 2π/3, so that T_I2 is inscribed in ABC. Triangle T_I2 is here named the 2nd Ehrmann inscribed triangle. T_I1 and T_I2 are inscribed in a common conic, here named the Ehrmann conic. The center of the Ehrmann conic is X₁₄₉₉₃.

  The orthocenter of T_C1 is the circumcenter of T_I1, and is here introduced (and at Bicentric Pairs (BicentricPairs.html)) as P(173). P(173) is therefore the intersection of the Euler lines of T_C1 and T_I1. The orthocenter of T_C2 is the circumcenter of T_I2, and is here introduced as U(173). U(173) is therefore the intersection of the Euler lines of T_C2 and T_I2. PU(173) lie on the Hatzipolakis axis (line PU(5) or X₅X₅₂₃). The midpoint of PU(173) is X₅. P(173), X₃ and X₄ are the vertices of an equilateral triangle with center P(5). U(173), X₃ and X₄ are the vertices of an equilateral triangle with center U(5).

  The vertex-triangle of T_C1 and T_C2 is here named the Ehrmann vertex-triangle. The Ehrmann vertex-triangle is the Kosnita triangle of the Johnson triangle (or equivalently, the reflection of the Kosnita triangle in X₅), and the tangential triangle of the Ehrmann mid-triangle (defined below). The vertex-triangle of T_I1 and T_I2 is ABC.

  Let V_AV_BV_C be the Ehrmann vertex-triangle. Then V_A is the isogonal conjugate of A'₁ wrt T_C1, and the isogonal conjugate of A'₂ wrt T_C2, and cyclically for V_B and V_C.

  The A-vertex of the Ehrmann vertex-triangle has barycentric coordinates:

  V_A = a^6 - a^4(b^2 + c^2) - a^2(b^4 + c^4) + (b^2 - c^2)^2(b^2 + c^2) : -(a^2 + b^2 - c^2)[(a^2 - b^2 - c^2)^2 - b^2c^2] : -(a^2 - b^2 + c^2)[(a^2 - b^2 - c^2)^2 - b^2c^2]

  The Ehrmann vertex-triangle is orthologic to ABC with orthology center X₄.

  The Ehrmann vertex-triangle is Eulerologic to ABC at X₇₅₇₇.

  The Ehrmann vertex-triangle is perspective to the reflections-of-P triangle for all P.

• X₁₈₅₈₈–X₁₈₇₅₂ — Collineations inverse-images

  contributed by César Eliud Lozada, May 4, 2018. Suppose that m is a collination. If P = m(Q), then P is the m collineation-image of Q, as in the preambles just before X₁₆₂₈₆ and X₁₆₅₀₄, and Q is here named the m inverse colllineation-image of P. Explicitly, if P = (A',B',C',U; A″,B″,C″V) collineation image of Q, then Q = (A″,B″,C″,V; A',B',c”,U) collination inverse-image of P.

• X₁₈₈₁₀–X₁₈₈₃₁ — Perspectors of Inconics

  Based on notes from Thanh Oai Dao, May 17, 2018. Let P be a point in the plane of a triangle ABC. Let

  A' = reflection of P in BC, and define B' and C' cyclically;

  Q = circumcenter of A'B'C' = isogonal conjugate of P;

  A″ = QA'∩BC, and define B″ and C″ cyclically.

  The triangle A″B″C″ is perspective to ABC. Let D(P) denote the perspector. The points P and Q are the foci of the inconic tangent to BC, CA, AB at A″, B″, C″, respectively.

  If P =\(p : q : r\) (barycentrics), then D(P) = q r/(\(b^{2}\)\(r^{2}\)+ \(c^{2}\)\(q^{2}\) + (\(b^{2}\) - \(c^{2}\) - \(a^{2})\) q r) : : , and

  D(P) = isotomic conjugate of the anticomplement of (midpoint of P and isogonal conjugate of P)

  D(P) = cevapoint of P and the orthocorrespondent of P

  D(P) = cross conjugate of (midpoint of P and isogonal conjugate of P) and G

  If P is on the infinity line, then D(P) = isotomic conjugate of P

  If P on the circumcircle or infinity line, then D(P) lies on the Steiner circumellipse.

  (Peter Moses, May 21, 2018)

• X₁₈₈₄₀–X₁₈₈₅₅ — Dao-bipedal-perspectors

  contributed by César Eliud Lozada, May 26, 2018. Let ABC be a triangle, P, Q two isogonal conjugate points and P_aP_bP_c, Q_aQ_bQ_c their respective pedal triangles. Let t be a real number, P'_a the point on PP_a such that PP'_a/PP_a=t and Q'_a the point on QQ_a such that QQ'_a/QQ_a=t; define P'_b, P'_c, Q'_b, Q'_c cyclically. Denote A'=PQ'_a ∩ QP'_a and similarly B' and C'. Then A'B'C' and ABC are perspective and, for given P, Q, the locus of the perspectors is a rectangular hyperbola. (Dao Thanh Oai, May 22, 2018)

• X₁₈₈₅₉–X₁₈₈₆₄ — Circumperp conjugates

  contributed by César Eliud Lozada, May 28, 2018. Let ABC be a triangle and P a point. The perpendicular bisectors of BC, CA, AB intersect the circumcircle at (A₁, A₂), (B₁, B₂), (C₁, C₂) ,respectively. Then the circumcircles of PA₁A₂, PB₁B₂, PC₁C₂ are coaxial. (Antreas Hatzipolakis, May 26, 2018, Anopolis 7568)

  Let Q be the point of intersection (other than P) of the three indicated circles. Then:

  Q = reflection-in-X₃ of (the circumcircle-inverse-of-P) Q = circumcircle-inverse-of-(reflection-of-P in X₃)

  For P other than X₃, the mapping P → Q is a conjugacy. The point Q is here named the Q=circumperp conjugate of P (as A₁, A₂, B₁, B₂, C₁, C₂ are the vertices of the circumperp triangles). For P=\(u:v:w\) (trilinears), coordinates of Q(P) are:

  Q(P) = (-a*(-b*c*SC*SB*u^2+2*w*v*SA^3)-SA*a*b*c*(b^2*v^2+c^2*w^2)+b*((SA+SB)*S^2-2*SA*SB^2)*w*u+c*((SA+SC)*S^2-2*SA*SC^2)*u*v)*a : :

  Some properties: +

  Q(P) lies on the line X₃P If P lies on the circumcircle, then Q(P) = reflection of P in X₃, i.e, Q(P) is the antipode of P in the circumcircle. If P lies on the line at infinity, then Q(P) = X₃

• X₁₈₉₀₉ — Anti-triangles

  contributed by César Eliud Lozada, June 4, 2018. Let T=A_tB_tC_t be a triangle perspective and orthologic to ABC. Suppose P_t is the perspector (ABC, T) and O_t is the orthologic center ABC to T, both expressed with respect to T. The anti-triangle-of-T is ABC and, if T is taken as the reference triangle, then its anti-triangle T'=A'_tB'_tC'_t is given by:

  A'_t = P_tA ∩ (perpendicular to BC through O_t)

  and cyclically for B'_t and C'_t.

• X₁₉₃₄₇–X₁₉₅₁₁ — Hutson triangles and related centers

  contributed by César Eliud Lozada, June 10, 2018. The following triangles are referenced in ETC by Randy Hutson: AAOA and AOA (in the preamble of X₁₅₀₁₅ and 2nd anti-extouch, anti-tangential-midarc and Lucas(±1) antipodal tangents (at the preamble of X₁₈₃₀₀). For definitions of these triangles, see index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html). Centers X₁₉₃₄₇-X₁₉₅₁₁ are perspectors, homothetic centers, orthologic centers and parallelogic centers of these triangles and other triangles. For a complete list, see X₁₉₃₄₇-Hutson-triangles.pdf (X(19347)-Hutson-triangles.pdf).

• X₁₉₉₀₁–X₁₉₉₂₆ — Polar co-centers

  contributed by César Eliud Lozada, June 26, 2018. Let T₁=A₁B₁C₁ and T₂=A₂B₂C₂ be two triangles not inscribed in the same circle and such that the perpendicular bisectors of A₁A₂, B₁B₂ and C₁C₂ concur at a point O. The point O provides a common center for handling T₁ and T₂ in polar coordinates. In such cases, triangles T₁ and T₂ are called polar co-centric and the point O is here named the polar co-center of T₁ and T₂.

Part 11

• X₂₁₁₅₃–X₂₁₁₆₈ — Gibert-Moses Centroids

  contributed by Peter Moses, August 20, 2018. Suppose that P =\(p : q : r\) (barycentrics) is a point in the plane of a triangle ABC, not on one of the sidelines BC, CA, AB. Let pK(K, P) denote the pivotal isocubic with pole K and pivot P (see Bernard Gibert’s Notations. The cubic pK(K,P) intersects the circumcircle in A, B, C and three other points, Q₁, Q₂, Q₃. The centroid of {Q₁, Q₂, Q₃} is the point, here named the P-Gibert-Moses centroid, given by the combo G(P) = 2*X₃ + P and

  \(G(P) = (b^{2} + c^{2} - a^{2)}(3a^{4} + b^{4} + c^{4} - 4 a^{2} b^{2} - 4 a^{2} c^{2} - 2 b^{2} c^{2})p - 2 a^{2} (q + r)\): :

• X₂₁₄₄₆–X₂₁₄₆₉ — Tripoles of axes of perspectivities

  contributed by César Eliud Lozada, August 25, 2018.

• X₂₁₅₀₉–X₂₁₅₄₃ — Brocard-Euler points of type 1

  Suppose that X is a point in the plane of a triangle ABC and that X has barycentrics

  \(a^{2}(h a^{2} + k b^{2} + k c^{2})\) : \(b^{2}(h b^{2} + k c^{2} + k a^{2})\) : \(c^{2}(h c^{2} + k a^{2} + k b^{2})\),

  where h and k are symmetric functions of (a,b,c) having the same degree of homogeneity. Then X is a triangle center that lies on the Brocard axis, X₃X₆, and the point

  (X₃,X₆,X₁,X₂;X₃,X₂,X₁,X₆) COLLINEATION IMAGE OF X

  lies on the Euler line of ABC. This point is here named the (h,k) Brocard-Euler point of type 1. (For type 2, see the preamble just before X₂₁₅₄₄.)

• X₂₁₅₄₄–X₂₁₅₇₇ — Brocard-Euler points of type 2

  Suppose that X is a point in the plane of a triangle ABC and that X has barycentrics

  \(a(h \cos A + k \sin A) : b(h \cos B + k \sin B) : c(h \cos C + k \sin C)\)

  where h and k are symmetric functions of (a,b,c) having the same degree of homogeneity. Then X is a triangle center that lies on the Brocard axis, X₃X₆, and the point

  (X₃,X₆,X₁,X₂;X₃,X₂,X₁,X₆) COLLINEATION IMAGE OF X

  lies on the Euler line of ABC. This point is here named the (h,k) Brocard-Euler point of type 2. (For type 1, see the preamble just before X₂₁₅₀₉.)

• X₂₁₆₁₆–X₂₁₆₆₃ — Wasat and anti-Wasat triangles

  contributed by César Eliud Lozada, August 27, 2018. Let ABC be a triangle, A'B'C' its incentral triangle and (I) its incircle. Denote (O_a) the circle with diameter AA' and \(r_{a}\) the radical axis of (O_a) and (I) and cyclically (O_b), (O_c) and \(r_{b}\), \(r_{c}\). Let A*B*C* be the triangle bounded by \(r_{a}\), \(r_{b}\), \(r_{c}\). Then: 1) ABC and A*B*C* share the same nine-point-center; 2) the medial triangle of ABC is the orthic triangle of A*B*C*; and 3) the medial triangle of A*B*C* is the 3rd Euler triangle of ABC. (See Antreas Hatzipolakis, Hyacinthos #28081). The triangle A*B*C* is here named the Wasat triangle of ABC. For Wasat triangle:

  The Wasat triangle is also the extraversion triangle of X₁₀, the complement of the excentral triangle, the reflection of the 2nd circumperp triangle in X₁₁₂₅, and the mid-triangle of the Ursa-minor and Ursa-major triangles. Also, the anti-Wasat triangle is also the anticomplementary triangle of the orthic triangle, and the reflection of the circumorthic triangle in X₃₈₉. (Randy Hutson, August 29, 2018)

  A* = b+c : c-a : b-a (barycentrics) |B*C*| = 2*R*cos(A/2) Area(A*B*C*) = (R*/r)*Area(ABC)/2

Part 12

• X₂₂₄₆₆–X₂₃₀₄₇ — Miscellaneous centers

  contributed by César Eliud Lozada, September 11, 2018.

• X₂₃₀₉₇–X₂₃₁₁₀ — K244 Moses images

  contributed by Randy Hutson, November 30, 2018. If a point P on the circumcircle of a triangle ABC has barycentrics \(p : q : r\), then then point a^2 q r (c^2 q + b^2 r) : : lies on the cubic K244. The following fourteen examples of K244 Moses images were contributed by Peter Moses, September 13, 2018. See also the preamble just before X₂₃₃₄₂. The Moses K244 image of P is the trilinear cube of the isogonal conjugate of P.

• X₂₂₂₉₁–X₂₃₃₃₃ — Anti-Ursa-minor triangle and related centers

  contributed by César Eliud Lozada, September 17, 2018. The anti-Ursa-minor triangle of an acute triangle ABC is the triangle A'B'C' whose Ursa-minor triangle is ABC. This triangle A'B'C' can be constructed as the anti-Hutson-intouch triangle of the Euler triangle of ABC, and A'B'C' has the following properties:

  A' = b^2+c^2 : c^2-a^2 : b^2-a^2 (barycentrics) |B'C'| = (a/4)*|sec(B)*sec(C)| area(A'B'C') = |sec(A)*sec(B)*sec(C)|*area(ABC)/8

• X₂₃₃₄₂–X₂₃₃₅₁ — K229 Moses images

  Let P be a point of the circumcircle of a triangle ABC, and let Q be the trilinear pole of the line X₆P. Let X be the Q-Ceva conjugate of P, so that also, X = crosspoint of P and Q. Then X lies on the cubic K229. See also the preamble just before X₂₃₀₉₇.

• X₂₃₃₅₂–X₂₃₃₅₅ — K635 Moses images

  Let P be a point of the circumconic with perspector X₁; that is, the circumconic with barycentric equation a y z + b z x + c x y = 0. Points on this conic include X_i for i = 88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799, 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599, 4604, 4606, 4607, 8052, 20332, as well as vertices of the Honsberger triangle (see X₇₆₇₀ and the inner Conway triangle (see X₁₁₆₇₇. Let Q be the trilinear pole of the line X₆P, and let X be the Q-Ceva conjugate of P, so that also, X = crosspoint of P and Q. Then X lies on the cubic K635. See also the preambles just before X₂₃₀₉₇ and X₂₃₃₄₂.

• X₂₃₅₈₂–X₂₃₅₉₄ — Points associated with barycentric squares of lines (inscribed ellipses)

  Several weeks before the appearance of this preamble on September 29, 2018, Clark Kimberling and Peter Moses, and independently, Randy Hutson, developed the notion of (pointwise) squares and cubes of points on lines, and they computed examples found below. Centers X₂₃₅₈₂-X₂₃₅₉₄ were contributed by Kimberling and Moses, and X₂₃₉₆₂-X₂₄₀₄₁ by Hutson. The latter are in two groups: X₂₃₉₆₂-X₂₃₉₉₂ (barycentrc squares) and X₂₃₉₉₃-X₂₄₀₄₁ (trilinear) See the preambles just before X₂₃₉₆₂ and X₂₃₉₉₃.

  Suppose that P =\(p : q : r\) (barycentrics) is a point in the plane of a triangle ABC, not on a sideline BC, CA, AB. Let L be the trilinear polar of P, so that L meets the sidelines in \(0 : q : -r\), \(-p : 0 : r\),\(p : -q : 0\). The barycentric squares of these points are the points A' = 0 : q^2 : r^2, B' = p^\(2 : 0 : r\)^2, C' = p^2 : q^2 : 0. The perspector of A'B'C' and ABC is the barycentric square P^2 = p^2 : q^2 : r^2, so that P^2 is the perspector of the inellipse that is the locus of squares of points on L. The center of the ellipse is the point p^2 (q^2 + r^2) : q^2 (r^2 + p^2) : r^2 (p^2 + q^2), which is the X₂-crosspoint of P^2, as well as the complement of the isotomic conjugate of P^2. The ellipse is here named the barycentric square of L.

• X₂₃₆₀₆–X₂₃₆₁₆ — Points associated with pointwise barycentric cubes of lines

  contributed by Clark Kimberling and Peter Moses, September 23, 2018. Suppose that P =\(p : q : r\) (barycentrics) is a point in the plane of a triangle ABC, not on a sideline BC, CA, AB. Let L be the trilinear polar of P, and let X be a point on L. The locus of a point X^3 (barycentric cube) is a cubic curve, here named the pointwise barycentric cubic of L. Let \(f(p,q,r,x,y,z) = q^{9}r^{9}x^{3} + 3p^{9}q^{3}r^{3}yz(q^{3}z+r^{3}y)\). An equation for the cubic follows:

  f(p,q,r,x,y,z) + f(q,r,p,y,z,x) + f(r,p,q,z,x,y) - 21 (p q r)³x y z = 0.

  An equivalent equation is (p^3 x + q^3 y + r^3 z)^3 - 27 p^3 q^3 r^3 x y z = 0.

  If L = X₂X₃ (the Euler line), then P = X₆₄₈, and the cubic passes through the points X_i for i = 2, 3081, 6524, 23606, 23607, 23608, 23609.

  If L is the line at infinity, then P = X₂, and the cubic is K656, which passes through X_i for i = 2, 3081, 6545, 8017, 8028, 8029, 8030, 8031, 8032, 23610, 23611, 23612, 23613, 23614, 23615, 23616.

• X₂₃₈₇₀–X₂₃₈₈₈ — Points on the infinity line

  contributed by Clark Kimberling and Peter Moses, September 27, 2018. Suppose that a point in the plane of a triangle ABC is given by P =\(p : q : r\) (barycentrics). The infinite difference point of P is introduced here as the point given by D(P) = q - r : r - p : p - q.

• X₂₃₉₆₂–X₂₃₉₉₂ — Points associated with barycentric squares of lines (inscribed ellipses)

  contributed by Randy Hutson, September 29, 2018 Continuing from the preamble before X₂₃₅₈₂, let L be a line in the plane of ABC and P =\(p : q : r\) (barycentrics) be the trilinear pole of L. Let E be the inellipse that is the (pointwise) barycentric square of L. The vertex conjugate of the foci of E is the point a^2 p^2 : b^2 q^2 : c^2 r^2. Let U =\(u : v : w\) and X =\(x : y : z\) be points on L so that U^2 and X^2 lie on E. The trilinear pole of the tangent to E at U^2 is the barycentric product P*U = p*u : q*v : r*w, which lies on the trilinear polar of the Brianchon point (perspector) of E. The intersection of the tangents to E at U^2 and X^2 is the barycentric product U*X = u*x : v*y : w*z.

• X₂₃₉₉₃–X₂₄₀₄₁ — Points associated with trilinear squares of lines (inscribed ellipses)

  contributed by Randy Hutson, September 29, 2018. The following discussion is analogous to the preambles just before X₂₃₅₈₂ and X₂₃₉₆₂. Suppose that P =\(p : q : r\) (trilinears) is a point in the plane of a triangle ABC, not on a sideline BC, CA, AB. Let L be the trilinear polar of P, so that L meets the sidelines in \(0 : q : -r\), \(-p : 0 : r\),\(p : -q : 0\). The trilinear squares of these points are the points A' = 0 : q^2 : r^2, B' = p^\(2 : 0 : r\)^2, C' = p^2 : q^2 : 0. The perspector of A'B'C' is the trilinear square P^2, so that P^2 = p^2 : q^2 : r^2 is the perspector of the inellipse that is the locus of squares of points on L. The center of the ellipse is the point p^2(q^2 + r^2) : q^2(r^2 + p^2) : r^2(p^2 + q^2), which is the X₂-crosspoint of P^2, as well as the complement of the isotomic conjugate of P^2. The ellipse is here named the trilinear square of L.

  Let E be the inellipse that is the trilinear square of L. The vertex conjugate of the foci of E is the point a^2 p^2 : b^2 q^2 : c^2 r^2. Let U =\(u : v : w\) and X =\(x : y : z\) be points on L so that U^2 and X^2 lie on E. The trilinear pole of the tangent to E at U^2 is the trilinear product P*U = p*u : q*v : r*w, which lies on the trilinear polar of the Brianchon point (perspector) of E. The intersection of the tangents to E at U^2 and X^2 is the trilinear product U*X = u*x : v*y : w*z.

Part 13

• X₂₄₃₀₇–X₂₄₃₀₉ — Perspectors involving obverse triangles

  contributed by Clark Kimberling, October 3, 2018. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. The obverse triangle of P is here introduced as the triangle A'B'C', where A' =\(p : r : q\), B' =\(r : q : p\), C' =\(q : p : r\). If P is a triangle center, then the obverse triangle of P is a central triangle.

• X₂₄₃₁₀–X₂₄₃₁₂ — Perspectors involving N-obverse triangles

  contributed by Clark Kimberling, October 3, 2018.

  Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. The N-obverse triangle of P is here introduced as the triangle A'B'C', where A' = \(- p : r : q\), B' =\(r : - q : p\), C' =\(q : p : - r\). If P is a triangle center, then the obverse triangle of P is a central triangle.

• X₂₄₄₇₇–X₂₄₄₈₁ — Centers associated with the obverse and N-obverse triangles of X₁

  contributed by Randy Hutson, October 5, 2018. Obverse and N-obverse triangles are introduced in preambles just before X₂₄₃₀₇ and X₂₄₃₁₀.

• X₂₄₄₈₂–X₂₄₅₁₈ — Collineation mappings involving trilinear obverse triangles

  contributed by Clark Kimberling, October 5, 2018. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. Trilinears for P are \(p/a : q/b : r/c\). The trilinear obverse triangle of P is here introduced as the triangle A'B'C', where A' =\(p/a : r/c : q/b\), B' =\(r/c : q/b : p/a\), C' =\(q/b : p/a : r/c\) (trilinears). The same triangle is given by barycentrics A' = b c p : b^2 r : c^2 q, B' = a^2 r : c a q : p c^2 p, C' = a^2 q : b^2 p : a b r. For example, the trilinear obverse triangle of X₂ has first barycentric b c : b^2 : c^2.

• X₂₄₅₁₉–X₂₄₅₃₆ — Collineation mappings involving trilinear N-obverse triangles

  contributed by Clark Kimberling, October 5, 2018., contributed by Clark Kimberling, October 5, 2018.Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. Trilinears for P are \(p/a : q/b : r/c\). The trilinear N-obverse triangle of P is here introduced as the triangle A'B'C', where A' = \(- p/a : r/c : q/b\), B' =\(r/c : - q/b : p/a\), C' =\(q/b : p/a : - r/c\) (trilinears). The same triangle is given by barycentrics A' = - b c p : b^2 r : c^2 q, B' = a^2 r : - c a q : p c^2 p, C' = a^2 q : b^2 p : - a b r. For example, the trilinear obverse triangle of X₂ has first barycentric - b c : b^2 : c^2.

• X₂₄₅₃₇–X₂₄₅₇₁ — Collineation mappings involving Gemini triangle 1

  contributed by Clark Kimberling, October 6, 2018.

  Following is a list of central triangles, by barycentric coordinates of A-vertex. The full names are Gemini triangle 1, Gemini triangle 2, Gemini triangle 3, etc. (Gemini is a constellation of stars - and the Latin noun for twin.) These triangles are cited in the sequel.

  Gemini 1 a : b + c : b + c

  Gemini 2 - a : b + c : b + c
  Gemini 3 a : a + b : a + c

  Gemini 4 - a : a + b : a + c
  Gemini 5 a : a - b : a - c

  Gemini 6 a : b - a : c - a
  Gemini 7 a : a - c : a - b

  Gemini 8 a : c - a : b - a (Garcia reflection triangle)
  Gemini 9 a : b + c - a : b + c - a

  Gemini 10 - a : b + c - a : b + c - a
  Gemini 11 a : a + b + c : a + b + c

  Gemini 12 - a : a + b + c : a + b + c
  Gemini 13 b +\(c : a : a\)

  Gemini 14 b +\(c : 2a : 2a\)
  Gemini 15 b +\(c : b : c\) (Gergonne line extraversion triangle; see X₁₀₁₈₀. See note below.)

  Gemini 16 b +\(c : c : b\) (See note below.)
  Gemini 17 b + c : b - c : c - b (See note below.)

  Gemini 18 b + c : c - b : b - c (See note below.)
  Gemini 19 b + c : a + b : a + c (See note below.)

  Gemini 20 2b +\(2c : a : a\) (See note below.)
  Gemini 21 a + b +\(c : a : a\)

  Gemini 22 a + b +\(c : - a : - a\)
  Gemini 23 a + b + c : b + c : b + c

  Gemini 24 a + b + c : - b - c : - b - c
  Gemini 25 a + b + c : a + b : a + c

  Gemini 26 a + b + c : a + c : a + b
  Gemini 27 a - b \(- c : a : a\)

  Gemini 28 a - b - c : b + c : b + c (See note below.)
  Gemini 29 a : b - c : c - b (See note below.)

  Gemini 30 a : c - b : b - c (Inner Conway triangle; see note below.)
  Gemini 31 b c : a^2 : a^2

  Gemini 32 - b c : a^2 : a^2
  Gemini 33 a^2 : b c : b c

  Gemini 34 - a^2 : b c : b c
  Gemini 35 cos \(A : 1 : 1\)

  Gemini 36 - cos \(A : 1 : 1\)
  Gemini 37 sec \(A : 1 : 1\)

  Gemini 38 - sec \(A : 1 : 1\)
  Gemini 39 -a + b + c : a + b + c : a + b + c

  Gemini 40 a + b + c : - a + b + c : - a + b + c
  Notes: (Randy Hutson, beginning November 9, 2018)

  (Gemini triangle 15) = complement of cevian triangle of X₇₅

  (Gemini triangle 16) = complement of incentral triangle

  (Gemini triangle 17) = anticomplement of cevian triangle of X₇₅

  (Gemini triangle 17) = anticomplement of anticomplement of Gemini triangle 15

  (Gemini triangle 18) = anticomplement of incentral triangle

  (Gemini triangle 19) = medial triangle of obverse triangle of X₁

  (Gemini triangle 19) = obverse triangle of X₁₀

  (Gemini triangle 20) = complement of Gemini triangle 28

  (Gemini triangle 23) = complement of Gemini triangle 39

  (Gemini triangle 28) = anticomplement of Gemini triangle 20

  (Gemini triangle 29) = anticomplement of extouch triangle

  (Gemini triangle 29) = anticomplement of anticomplement of 1st Zaniah triangle

  (Gemini triangle 30) = anticomplement of intouch triangle

  (Gemini triangle 30) = anticomplement of anticomplement of 2nd Zaniah triangle

  (Gemini triangle 33) = unary cofactor triangle of 2nd Sharygin triangle

  (Gemini triangle 34) = unary cofactor triangle of 1st Sharygin triangle

  (Gemini triangle 39) = anticomplement of Gemini triangle 23

  For more Gemini triangles, see the preambles just before X₂₆₁₅₃ and X₂₇₃₇₈.

  If T is a central triangle A'B'C' with A' of the form \(f(a,b,c) : g(a,b,c) : g(a,b,c)\), then the (A, B, C,X₂; A',B',C',X₂) collineation image of the Euler line is the Euler line. Examples include Gemini triangles 1,2,9,10,11,12,13,14,21,22,23,24,27,28, and 31-40.

  Let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 1, as in centers X₂₄₅₃₇-X₂₄₅₇₁. Then

  m(X) = a (a + b - c)(a - b + c) x + (b + c - a) (a + b - c) (a + c) y + (b + c - a) (a - b + c) (a + b) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₄₅₇₄–X₂₄₅₇₉ — Centers associated with the trilinear obverse and trilinear N-obverse triangles of X₂

  contributed by Randy Hutson, October 6, 2018. Obverse and N-obverse triangles are introduced in preambles just before X₂₄₄₈₂ and X₂₄₅₁₉.

• X₂₄₅₈₀–X₂₄₆₃₈ — Collineation mappings involving Gemini triangle 2

  contributed by Clark Kimberling, October 7, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 2, as in centers X₂₄₅₈₀-X₂₄₆₃₈. Then

  m(X) = a x - (a + c) y - (a + b) z : : , and m(X) is on the Euler line if and only if X is on the Euler line.

  Fixed points: m(X₂) = X₂, m(X₂₃₉)= X₂₃₉, m(X₆₄₉)= X₆₄₉

  Cycles: m(X₅₇) = X₆₃, m(X₆₃) = X₅₇, and m(X₈₈) = X₁₉₀, m(X₁₉₀) = X₈₈

• X₂₄₆₅₂–X₂₄₆₇₉ — Collineation mappings involving Gemini triangle 3

  contributed by Clark Kimberling, October 8, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 3, as in centers X₂₄₆₅₂-X₂₄₆₇₉. Then

  m(X) = a (a b + a c - b c) x + (a^2 b - a^2 c + a b ^2 + b^2 c) y + (a^2 c - a^2 b + a c ^2 + c^2 b) z : : .

• X₂₄₆₈₂–X₂₄₇₂₆ — Collineation mappings involving Gemini triangle 4

  contributed by Clark Kimberling, October 8, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 4, as in centers X₂₄₆₈₂-X₂₄₇₂₆. Then m(X) = a x - (a + b) y - (a + c) y : : .

• X₂₄₇₃₅–X₂₄₇₅₈ — Collineation mappings involving Gemini triangle 5

  contributed by Clark Kimberling, October 9, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 5, as in centers X₂₄₇₃₅-X₂₄₇₅₈. Then

  m(X) = a(a b + a c - b c) x + (a^2 c - a^2 b + a b^2 + b^2 c - 2 a b c) y + (a^2 b - a^2 c + a c^2 + b c^2 - 2 a b c) z : : .

• X₂₄₇₅₉–X₂₄₇₇₀ — Collineation mappings involving Gemini triangle 6

  contributed by Clark Kimberling, October 9, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 6, as in centers X₂₄₇₅₉-X₂₄₇₇₀. Then

  m(X) = a(a b + a c - 3 b c) x + (a^2 b - 3 a^2 c - a b^2 - b^2 c + 4 a b c) y + (a^2 c - 3 a^2 b - a c^2 -b c^2 + 4 a b c) z : : .

• X₂₄₇₇₃–X₂₄₇₉₅ — Collineation mappings involving Gemini triangle 7

  contributed by Clark Kimberling, October 9, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 7, as in centers X₂₄₇₇₃-X₂₄₇₉₅. Then

  m(X) = a^2 (a - b - c) x + b ( -a b + b^2 + a c - 2 b c + c^2) y + c (-a c + c^2 + a b - 2 b c + b^2) z : : .

• X₂₄₇₉₆–X₂₄₈₀₅ — Collineation mappings involving the Garcia reflection triangle (Gemini triangle 8)

  contributed by Clark Kimberling, October 9, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 8, as in centers X₂₄₇₉₆-X₂₄₈₀₅. Then

  m(X) = a (a^2 - a b + 2 b^2 - a c - 4 c b + 2c^2) x - (2a^b + a b^2 - b^3 + 2 a^2 c + 3 a b c + 2 b^2 c - 4 a c^2 - 3 b c^2 + 2 c^3) y - (2a^c + a c^2 - c^3 + 2 a^2 b + 3 a b c + 2 b c^2 - 4 a b^2 - 3 b^2 c + 2 b^3) z : :

• X₂₄₈₀₆–X₂₄₈₅₄ — Centers related to obverse triangles

  contributed by César Eliud Lozada, October 10, 2018. Obverse triangle of X₁ (defined just before of X24307): Perspective triangles and perspectors:

  (ABC, 75), (anti-tangential-midarc, 24806), (anticomplementary, 1654), (Aquila, 3679), (1st circumperp, 24309), (2nd circumperp, 993), (excentral, 9), (extangents, 24308), (2nd extouch, 9), (Fuhrmann, 10), (inner-Garcia, 1), (outer-Garcia, 3679), (Garcia-reflection, 9), (intangents, 24307), (medial, 9), (1st Sharygin, 8424), (2nd Sharygin, 4363), (tangential, 8424), (2nd Zaniah, 9), (N-obverse of X₁, 75), (trilinear obverse of X₂, 894),

  Orthologic triangles and orthologic centers:

  (anti-Artzt, 3679, 24807), (Artzt, 3679, 24808), (1st Parry, 1, 24809), (2nd Parry, 1, 24810), (1st tri-squares, 3679, 24811), (2nd tri-squares, 3679, 24812)

  Parallelogic triangles and parallelogic centers:

  (ABC, 1, 190), (ABC-X3 reflections, 1, 24813), (anti-Aquila, 1, 4432), (anti-Ara, 1, 24814), (5th anti-Brocard, 1, 24815), (2nd anti-circumperp-tangential, 1, 24816), (anti-Euler, 1, 24817), (anti-inner-Grebe, 1, 24818), (anti-outer-Grebe, 1, 24819), (anti-Mandart-incircle, 1, 24820), (anticomplementary, 1, 4440), (Aquila, 1, 24821), (Ara, 1, 24822), (1st Auriga, 1, 24823), (2nd Auriga, 1, 24824), (5th Brocard, 1, 24825), (2nd circumperp tangential, 1, 24826), (Ehrmann-mid, 1, 24827), (Euler, 1, 24828), (outer-Garcia, 1, 24715), (Gossard, 1, 24830), (inner-Grebe, 1, 24831), (outer-Grebe, 1, 24832), (Johnson, 1, 24833), (inner-Johnson, 1, 24834), (outer-Johnson, 1, 24835), (1st Johnson-Yff, 1, 24836), (2nd Johnson-Yff, 1, 24837), (Lucas homothetic, 1, 24838), (Lucas(-1) homothetic, 1, 24839), (Mandart-incircle, 1, 24840), (medial, 1, 1086), (5th mixtilinear, 1, 24841), (3rd tri-squares-central, 1, 24842), (4th tri-squares-central, 1, 24843), (X3-ABC reflections, 1, 24844), (Yff contact, 10, 190), (inner-Yff, 1, 24845), (outer-Yff, 1, 24846), (inner-Yff tangents, 1, 24847), (outer-Yff tangents, 1, 24848)

  N-obverse triangle of X₁ (defined just before of X24310) Perspective triangles and perspectors:

  (ABC, 75), (anti-tangential-midarc, 24849), (extangents, 24310), (5th extouch, 24312), (outer-Garcia, 1), (medial, 1), (1st Sharygin, 24311), (2nd Sharygin, 8301), (tangential, 8301), (1st Zaniah, 1), (obverse of X₁, 75), (trilinear N-obverse of X₂, 239)

  Orthologic triangles and orthologic centers:

  (excenters-midpoints, 1, 24850), (Garcia-reflection, 1, 24851), (2nd Schiffler, 1, 24852)

  Trilinear obverse triangle of X₂ (defined just before of X24482) Perspective triangles and perspectors:

  (ABC, 6), (anti-Conway, 6), (2nd anti-Conway, 6), (anti-inner-Grebe, 6), (anti-outer-Grebe, 6), (anti-Honsberger, 6), (anticomplementary, 192), (2nd Brocard, 6), (circumsymmedial, 6), (2nd Ehrmann, 6), (excentral, 1045), (9th Fermat-Dao, 6), (10th Fermat-Dao, 6), (13th Fermat-Dao, 6), (14th Fermat-Dao, 6), (inner-Grebe, 6), (outer-Grebe, 6), (incentral, 192), (1st Kenmotu diagonals, 6), (2nd Kenmotu diagonals, 6), (2nd mixtilinear, 24853), (2nd orthosymmedial, 6), (symmedial, 6), (tangential, 6), (inner tri-equilateral, 6), (outer tri-equilateral, 6), (obverse of X₁, 894), (trilinear N-obverse of X₂, 6)

  Orthologic triangles and orthologic centers:

  (1st Ehrmann,6,24854)

  Trilinear N-obverse triangle of X₂ (defined just before of X24519) Perspective triangles and perspectors:

  (ABC, 6), (anti-Conway, 6), (2nd anti-Conway, 6), (anti-inner-Grebe, 6), (anti-outer-Grebe, 6), (anti-Honsberger, 6), (2nd Brocard, 6), (circumsymmedial, 6), (2nd Ehrmann, 6), (9th Fermat-Dao, 6), (10th Fermat-Dao, 6), (13th Fermat-Dao, 6), (14th Fermat-Dao, 6), (inner-Grebe, 6), (outer-Grebe, 6), (incentral, 2), (1st Kenmotu diagonals, 6), (2nd Kenmotu diagonals, 6), (2nd orthosymmedial, 6), (symmedial, 6), (tangential, 6), (inner tri-equilateral, 6), (outer tri-equilateral, 6), (N-obverse of X₁, 239), trilinear obverse of X₂, 6)

  Orthologic triangles and orthologic centers:

  (1st Ehrmann,6,24829)

• X₂₄₈₆₄–X₂₄₈₇₉ — Collineation mappings involving Gemini triangle 9

  contributed by Clark Kimberling, October 10, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 9, as in centers X₂₄₈₆₄-X₂₄₈₇₉. Then

  m(X) = (a^3 - a^2 b - 2 a b^2 - a^2 c + 5 a b c - 2 a c^2) x + (- 2 a^3 + a^2 b + 2 a b^2 - b^3 + 3 a^2 c - 7 a b c + 2 b^2 c + 3 a c^2 + b c^2 - 2 c^3) y + (- 2 a^3 + a^2 c + 2 a c^2 - c^3 + 3 a^2 b - 7 a b c + 2 b c^2 + 3 a b^2 + b^2 c - 2 b^3) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₄₈₈₀–X₂₄₉₂₅ — Collineation mappings involving Gemini triangle 10

  contributed by Clark Kimberling, October 10, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 10, as in centers X₂₄₈₈₀-X₂₄₉₂₅. Then

  m(X) = a (a^2 + a b + a c + b c) x + (-a^2 b + b^2 - a^2 c - a b c - a c^2 - b c^2) y + (-a^2 c + c^2 - a^2 b - a b c - a b^2 - b^2 c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₄₉₃₁–X₂₄₉₆₂ — Collineation mappings involving Gemini triangle 11

  contributed by Clark Kimberling, October 11, 2018.Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 11, as in centers X₂₄₉₃₁-X₂₄₉₆₂. Then

  m(X) = a (a^2 + a b + a c + b c) x + (a^2 b + 2 a b^2 + b^3 + a^2 c + 3 a b c + 2 b^2 c + a c^2 + b c^2) y + (a^2 c + 2 a c^2 + c^3 + a^2 b + 3 a b c + 2 b c^2 + a b^2 + b^2 c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₄₉₇₃–X₂₄₉₈₀ — 4th intersections of inconics and the circumcircles of their polar triangles

  contributed by Randy Hutson, October 11, 2018. Let E be an inconic (other than the incircle) with Brianchon point (perspector) P. Let A'B'C' be the polar triangle of E, which is the cevian triangle of P. Let O* be the circumcircle of A'B'C' (i.e., the cevian circle of P). E and O* intersect in 4 points: A', B', C' and a 4th intersection which is a triangle center. The appearance of (E,i,O*,j) in the following list means that inconic E with perspector X_i intersects circle O* (other than at A', B', C') at X_j:

  (Brocard inellipse, 6, symmedial circle, 24973)

  (incentral inellipse, 1, incentral circle, 23063)

  (Kiepert parabola, 99, 2nd Steiner circle, 24974)

  (Lemoine inellipse, 598, 3rd Lemoine circle, 20383)

  (MacBeath inconic, 264, MacBeath circle, 24977)

  (Mandart inellipse, 8, Mandart circle, 11)

  (orthic inconic, 4, nine-point circle, 125)

  (Steiner inellipse, 2, nine-point circle, 115)

  (Yff parabola, 190, cevian circle of X₁₉₀, 24979)

• X₂₄₉₈₂–X₂₅₀₂₄ — Collineation mappings involving Gemini triangle 13

  contributed by Clark Kimberling, October 12, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 13, as in centers X₂₄₉₈₂-X₂₅₀₂₄. Then

  m(X) = (-a^2 b + b^3 - a^2 c - b^2 c - b c^2 + c^3) x + b(a^2 - b^2 - 2 a c + c ^2) y + c(a^2 - c^2 -2a b + b^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₅₀₂₅ — Collineation mappings involving Gemini triangle 14

  contributed by Clark Kimberling, October 12, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 13, as in centers X₂₅₀₂₅-X₂₅₀₄₁. Then

  m(X) = : : (a^2 b + a b^2 + 2 b^3 - a^2 c + 2 a b c - 3 b^2 c + a c^2 - 3 b c^2 + 2 c^3) x + (4 a^2 b + 2 a b^2 - 2 b^3 - 10 a b c + 2 b^2 c + 4 b c^2) y + (4 a^2 c - 10 a b c + 4 b^2 c + 2 a c^2 + 2 b c^2 - 2 c^3) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₅₀₄₅–X₂₅₀₅₀ — Points associated with a family of ellipses

  contributed by Peter Moses, October 10, 2018. Let E be the circumellipse with center X₁₄₁, given by

  a^2 (b^2 + c^2) y z + b^2 (c^2 + a^2) x z + c^2 (a^2 + b^2) x y = 0.

  This ellipse is the isogonal conjugate of the anticomplement of the de Longchamps line, and it passes through X_i for i = 67, 110, 660, 670, 694, 1634, 4553, 4576, 8050, 20021.

  For arbitrary fixed nonzero constant k, let E(k) be the ellipse obtained from E by dilation from X₁₄₁ with ratio k; then E(k) is given by

  (b^2 + c^2) ((a^2 + b^2) (a^2 + c^2) k^2 - b^2 c^2) x^2 + 2 (a^2 b^2 c^2 + (b^2 + c^2) (c^2 + a^2) (a^2 + b^2) k^2) y z + (cyclic) = 0.

  The major axis of every E(k) passes through X_i for i = 6, 1344, 2574, 8105, 9173, 13414, and the minor axis, through X_i for i = 6, 1345, 2575, 8106, 9174, 13415, 14899. The two axes are parallel to the Simson lines of X₁₁₁₃ and X₁₁₁₄.

  These ellipses are associated with a problem in navigation posed by William Lionheart (see the link at X₆), in connection with the fact that if a point P moves on E(k), the sum of squared distances from P to the sidelines BC, CA, AB stays constant. The value of the constant is

  (a^2 b^2 c^2 + (b^2 + c^2) (c^2 + a^2) (a^2 + b^2) k^2) / (4 (a^2 + b^2 + c^2) R^2).

  Suppose that P =\(p : q : r\) is a point on the circumcircle. Then the point

  LM(P,k) = (b^2 + c^2) k ((2 a^2 + b^2 + c^2) p + (-a^2 - c^2) q + (-a^2 - b^2) r) - a^2 ((b^2 + c^2) p + (c^2 + a^2) q + (a^2 + b^2) r) : :

  is on the ellipse E(k). The point LM(P,k) is here named the Lionheart-Moses k-image of P. Centers X₂₅₀₄₅-X₂₅₀₅₀ are examples of such images, for k = 1, as are X₃₄₄₈, X₁₁₀₆₁, X₂₅₃₃₂, and X₂₅₃₃₃; see also X₂₅₃₁₄-X₂₅₃₃₆.

• X₂₅₀₆₁–X₂₅₁₀₁ — Collineation mappings involving Gemini triangle 15

  contributed by Clark Kimberling, October 13, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 15 (the Gergonne line extraversion triangle; see X₁₀₁₈₀), as in centers X₂₅₀₆₁-X₂₅₁₀₀. Then

  m(X) = a [(b + c)(a - b - c) x + (b^2 - a b - b c) y + (c^2 - a c - b c) z]) : : .

• X₂₅₁₀₂–X₂₅₁₄₆ — Collineation mappings involving Gemini triangle 16

  contributed by Clark Kimberling, October 13, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 16, as in centers X₂₅₁₀₂-X₂₅₁₄₆. Then

  m(X) = (b + c)(a b + a c - b c) + (a b c - a c^2 + b^2 c) y + (a b c - a b^2 + b c^2) z : : .

• X₂₅₁₅₁–X₂₅₂₃₆ — Centers related to Fermat-Dao equilateral triangles

  contributed by César Eliud Lozada, October 13, 2018. For definitions and coordinates of all Fermat-Dao equilateral triangles, see Index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html#F). A complete list of orthologic and parallelogic triangles to Fermat-Dao triangles and centers can be downloaded from here (Fermat-Dao-centers.pdf).

• X₂₅₂₃₇–X₂₅₂₇₂ — Collineation mappings involving Gemini triangle 17

  contributed by Clark Kimberling, October 14, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 17, as in centers X₂₅₂₃₇-X₂₅₂₇₂. Then

  m(X) = a (b + c) (b + c - a) x + b (c - a)(c + a - b) y + c (b - a) (b + a - c) z : : .

• X₂₅₂₇₃–X₂₅₃₁₃ — Collineation mappings involving Gemini triangle 18

  contributed by Clark Kimberling, October 14, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 18, as in centers X₂₅₂₇₃-X₂₅₃₁₃. Then

  m(X) = (b + c) (b c + c a + a b) x + (c - a)(b c + c a + a b) y + ( (b - a) (b c + c a + a b) z : : .

• X₂₅₃₁₄–X₂₅₃₃₆ — Lionheart-Moses images

  as defined in the preamble just before X₂₅₀₄₅.

• X₂₅₃₄₁–X₂₅₃₈₅ — Collineation mappings involving Gemini triangle 19

  contributed by Clark Kimberling, October 15, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 19, as in centers X₂₅₃₄₁-X₂₅₃₈₅. Then

  m(X) = (b + c) x + (a + b) y + (a + c) z : : .

• X₂₅₃₈₆–X₂₅₄₀₀ — Collineation mappings involving Gemini triangle 20

  contributed by Clark Kimberling, October 15, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 20, as in centers X₂₅₃₈₆-X₂₅₄₀₀. Then

  m(X) = -2 (b + c) (2 a + 2 b - c)(2a - b + 2 c) x + b (a - 2 b - 2 c)(2 a + 2 b - c) y + c (a - 2 b - 2 c) (2 a + 2c - b) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₅₄₁₇–X₂₅₄₃₁ — Collineation mappings involving Gemini triangles 1 - 4

  contributed by Randy Hutson, October 17, 2018. Gemini triangles are introduced in the preamble just before X₂₄₅₃₇.

• X₂₅₄₄₁–X₂₅₄₇₃ — Collineation mappings involving Gemini triangle 21

  contributed by Clark Kimberling, October 19, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 21, as in centers X₂₅₄₄₁-X₂₅₄₇₃. Then

  m(X) = (a + b)(a + c)(a + b + c) x + b (a + b)(b + c) y + c (a + c)(b + c) z : : .

  A point X lies on the Euler line if and only m(X) lies on the Euler line.

• X₂₅₄₇₄–X₂₅₄₈₄ — Collineation mappings involving Gemini triangle 22

  contributed by Clark Kimberling, October 19, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 22, as in centers X₂₅₄₇₄-X₂₅₄₈₄. Then

  m(X) = (a + b + c)(a + 2b + c)(a + b + 2 c) x - b (2 a + b + c)(a + b + 2 c) y - c (2 a + b + c)(a + 2 b + c) z : : .

  A point X lies on the Euler line if and only m(X) lies on the Euler line.

• X₂₅₄₉₀–X₂₅₅₄₂ — Collineation mappings involving Gemini triangle 23

  contributed by Clark Kimberling, October 19, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 23, as in centers X₂₅₄₉₀-X₂₅₅₄₂. Then

  m(X) = b c (a + b + c) x + a c (a + c) y + a b ( a + b) z : : .

  Fixed points of m include X_i for these i: 2, 36, 238, 667. A point X lies on the Euler line if and only m(X) lies on the Euler line.

• X₂₅₅₄₃–X₂₅₅₅₄ — Collineation mappings involving Gemini triangle 24

  contributed by Clark Kimberling, October 19, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 24, as in centers X₂₅₅₄₃-X₂₅₅₅₄. Then

  m(X) = (a + b + c)(2a + b + c)(2a + b + 2c) x - (a + c)(2a + b + c) (a + 2b + 2c) y - (a + b)(2a + b + c)(a + 2b + 2c) : : .

  A point X lies on the Euler line if and only m(X) lies on the Euler line.

• X₂₅₅₆₇–X₂₅₅₈₀ — Collineation mappings involving Gemini triangles 3 - 6

  contributed by Randy Hutson, October 22, 2018. The Gemini triangles are introduced in the preamble just before X₂₄₅₃₇.

• X₂₅₅₈₁–X₂₅₆₀₅ — Collineation mappings involving Gemini triangle 25

  contributed by Clark Kimberling, October 23, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 25, as in centers X₂₅₅₈₁-X₂₅₆₀₅. Then

  m(X) = a(a + b + c)(a - b - c) x - b (a + b)(a - b + c) y - c (a + c)(a + b - c) z : : .

• X₂₅₆₁₀–X₂₅₆₃₈ — Collineation mappings involving Gemini triangle 26

  contributed by Clark Kimberling, October 23, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 26, as in centers X₂₅₆₁₀-X₂₅₆₃₈. Then

  m(X) = (a + b + c)(a b + a c - b c) x + (b + c)(a b - a c + b c) y + (b + c)(a c - a b + b c) z : : .

• X₂₅₆₄₅–X₂₅₆₈₉ — Collineation mappings involving Gemini triangle 27

  contributed by Clark Kimberling, October 24, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 27, as in centers X₂₅₆₄₅-X₂₅₆₈₉. Then

  m(X) = (a + b) (a + c)(a - b - c) x + b (b + c) (a + b) y + c (b + c) (a + c) z : : .

  The point m(X) is on the Euler line if and only if the point X is on the Euler line.

• X₂₅₆₉₀–X₂₅₇₁₀ — Collineation mappings involving Gemini triangle 28

  contributed by Clark Kimberling, October 24, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 28, as in centers X₂₅₆₉₀-X₂₅₇₁₀. Then

  m(X) = (a - b - c) (2 a + 2 b - c) (2 a - b + 2 c) x - (a + c)(a - 2 b - 2 c)(2 a + 2 b - c) y : (a + b)(a - 2 b - 2 c)(2 a + 2 c - b) z : : .

  The point m(X) is on the Euler line if and only if the point X is on the Euler line.

• X₂₅₇₁₆–X₂₅₇₂₆ — Collineation mappings involving Gemini triangle 29

  contributed by Clark Kimberling, October 25, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 29, as in centers X₂₅₇₁₆-X₂₅₇₂₆. Then

  m(X) = a(3 a^2 - b^2 - c^2 - 2 a b - 2 a c + 2 b c) x + (c - a) (a^2 - 3 b^2 + c^2 - 2 a b - 2 a c + 2 b c) y + (b - a) (a^2 - 3 c^2 + b^2 - 2 a c - 2 a b + 2 b c) z : : .

• X₂₅₇₂₇–X₂₅₇₃₇ — Collineation mappings involving Gemini triangle 30

  contributed by Clark Kimberling, October 25, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 30, as in centers X₂₅₇₂₇-X₂₅₇₃₇. Then

  m(X) = a (a - b - c) (3 a - b - c) x + (c - a) (a - 3 b + c) (a - b + c) y + (b - a) (a - 3c + b) (a - c + b) z : : .

• X₂₅₇₄₁–X₂₅₇₆₇ — Collineation mappings involving Gemini triangle 31

  contributed by Clark Kimberling, October 26, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 31, as in centers X₂₅₇₄₁-X₂₅₇₆₇. Then

  m(X) = b c (b ^2 - a c)(c^2 - a b) x + b^2 (a^2 - b c)(c^2 - a b) y + c^2 (a^2 - b c) (b^2 - a c) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₅₇₆₈–X₂₅₇₉₉ — Collineation mappings involving Gemini triangle 32

  contributed by Clark Kimberling, October 26, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 32, as in centers X₂₅₇₆₈-X₂₅₇₉₉. Then

  m(X) = b c (b ^2 + a c)(c^2 + a b) x - b^2 (a^2 + b c)(c^2 + a b) y - c^2 (a^2 + b c) (b^2 + a c) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₅₈₀₀–X₂₅₈₃₇ — Collineation mappings involving Gemini triangle 33

  contributed by Clark Kimberling, October 27, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 33, as in centers X₂₅₈₀₀-X₂₅₈₃₇. Then

  m(X) = a^2 (b^2 - a c) (c^2 - a b) x + a c (a^2 - b c) (c^2 - a b) y + a b (a^2 - b c) (b^2 - a c) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₅₈₃₈–X₂₅₈₇₄ — Collineation mappings involving Gemini triangle 34

  contributed by Clark Kimberling, October 27, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 34, as in centers X₂₅₈₃₈-X₂₅₈₇₄. Then

  m(X) = a^2 (b^2 + a c) (c^2 + a b) x - a c (a^2 + b c) (c^2 + a b) y - a b (a^2 + b c) (b^2 + a c) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₅₈₇₅–X₂₅₉₂₉ — Collineation mappings involving Gemini triangle 35

  contributed by Clark Kimberling, October 28, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 35, as in centers X₂₅₈₇₅-X₂₅₉₂₉. Then

  m(X) = (a - b - c)(a^2 - b^2 - c^2) x + 2 a c (a - b + c) y + 2 a b (a - c + b) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₅₉₃₀–X₂₅₉₅₆ — Collineation mappings involving Gemini triangle 36

  contributed by Clark Kimberling, October 28, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 36, as in centers X₂₅₉₃₀-X₂₅₉₅₆. Then

  m(X) = (a + b - c) (a - b + c) (a^2 - b^2 - c^2) x + 2 a c (b + c - a) (a + b - c) y + 2 a b (b + c - a) (a - b + c) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₅₉₅₇–X₂₅₉₆₁ — Points with 1st barcentric b3 + k a b c + c3

  contributed by Peter Moses, October 28, 2018. For each real number k, let P(k) = a (a^2 - b c) ((a^2 - b c) (a b^3 + a c^3 - 2 b^2 c^2) - b c (a c - b^2) (a b - c^2) k) : : . Then the point (A, B, C,X₂; A',B',C',X₂) collineation image of P(k), where A'B'C' = Gemini triangle 31, is the point with barycentrics b^3 + k a b c + c^3 : : . Both P(k) and Q(k) are on the line X₂X₃₁.

• X₂₅₉₆₂–X₂₅₉₉₇ — Collineation mappings involving Gemini triangle 37

  contributed by Clark Kimberling, October 29, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 37, as in centers X₂₅₉₆₂-X₂₅₉₉₇. Then

  m(X) = 2 b c (b + c - a) x + (a - b + c) (a^2 - b^2 + c^2) y + (a + b - c) (a^2 + b^2 - c^2) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line; examples: m(X₄) = X₁₁₁₀₉ and X₂₃ = X₁₁₂₈₅.

Part 14

• X₂₆₀₀₁–X₂₆₀₂₆ — Collineation mappings involving Gemini triangle 38

  contributed by Clark Kimberling, October 29, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 38, as in centers X₂₆₀₀₁-X₂₆₀₂₆. Then

  m(X) = 2 b c (a - b + c) (a + b - c) x + (a - b - c) (a^2 + b^2 - c^2) y + (a - b - c) (a^2 - b^2 + c^2) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₆₀₂₇–X₂₆₀₆₄ — Collineation mappings involving Gemini triangle 39

  contributed by Clark Kimberling, October 29, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 39, as in centers X₂₆₀₂₇-X₂₆₀₈₄. Then

  m(X) = 2 b c (a - b - c) x - a c(a + b + c) y - a b (a + b + c) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₆₀₉₁–X₂₆₁₅₂ — Collineation mappings involving Gemini triangle 40

  contributed by Clark Kimberling, October 29, 2018. Extending the preamble just before X₂₄₅₃₇, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 40, as in centers X₂₆₀₉₁-X₂₆₁₅₂. Then

  m(X) = b c (a + b + c) x + a c (a - b + c) y + a b (a + b - c) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₆₁₅₃–X₂₆₁₈₀ — Collineation mappings involving Gemini triangle 41

  contributed by Clark Kimberling, October 30, 2018.Following is a list of central triangles, by barycentric coordinates of A-vertex. The full names are Gemini triangle 41, Gemini triangle 42, Gemini triangle 43, etc. See the preamble just before X₂₄₅₃₇ for the definitions of Gemini triangles 1-40.

  Gemini 41 b^2 + c^2 : a^2 : a^2

  Gemini 42 a^2 + b^2 + c^2 : a^2 : a^2
  Gemini 43 a^2 : b^2 + c^2 : b^2 + c^2

  Gemini 44 - a^2 : b^2 + c^2 : b^2 + c^2 (circum-medial triangle, TCCT 6.19
  Gemini 45 (b - c)^2 : a^2 : a^2

  Gemini 46 (b + c)^2 : a^2 : a^2
  Gemini 47 a^2 : (b + c)^2 : (b + c)^2

  Gemini 48 a^2 : (b - c)^2 : (b - c)^2
  Gemini 49 (b + c)^2 : (b - c)^2 : (b - c)^2

  Gemini 50 (b - c)^2 : (b + c)^2 : (b + c)^2
  Gemini 51 (b - c)^2 : b^2 + c^2 : b^2 + c^2

  Gemini 52 (b + c)^2 : b^2 + c^2 : b^2 + c^2
  Gemini 53 b^2 + c^2 : (b - c)^2 : (b - c)^2

  Gemini 54 b^2 + c^2 : (b + c)^2 : (b + c)^2
  Gemini 55 a^2 : 2 b c : 2 b c

  Gemini 56 - a^2 : 2 b c : 2 b c
  Gemini 57 b^2 + c^2 : b c : b c

  Gemini 58 b^2 + c^2 : - b c : - b c
  Gemini 59 - b c + c a + a b : b c + c a + a b : b c + c a + a b

  Gemini 60 b c + c a + a b : - b c + c a + a b : - b c + c a + a b

  If T is a central triangle A'B'C' with A' of the form \(f(a,b,c) : g(a,b,c) : g(a,b,c)\), then the (A, B, C,X₂; A',B',C',X₂) collineation image of the Euler line is the Euler line. Examples include Gemini triangles 30-60.

  Let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 41, as in centers X₂₆₁₅₃-X₂₆₁₈₀. Then

  m(X) = (a^2 - b^2 + c^2) (a^2 + b^2 - c^2) (b^2 + c^2 ) x + (b^2 (b^2 + c^2 - a^2) ( axxx : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₆₁₈₁–X₂₆₁₉₉ — Collineation mappings involving Gemini triangle 42

  contributed by Clark Kimberling, October 30, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 42, as in centers X₂₆₁₈₁-X₂₆₁₉₉. Then

  m(X) = (a^2 + b^2) (a^2 + c^2) (a^2 + b^2 + c^2)x + b^2 (a^2 + b^2) (b^2 + c^2) y + c^2 (a^2 + c^2) (b^2 + c^2) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₆₂₀₃–X₂₆₂₂₆ — Collineation mappings involving Gemini triangle 43

  contributed by Clark Kimberling, October 31, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 43, as in centers X₂₆₂₀₃-X₂₆₂₂₆. Then

  m(X) = a^2 (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) x + (a^2 + c^2) (b^2 + c^2 - a^2) (a^2 + b^2 - c^2) y + (a^2 + b^2) (b^2 + c^2 - a^2) (a^2 - b^2 + c^2) z : : .

  A point X lies on the Euler line if and only if m(X) also lies on the Euler line.

• X₂₆₂₂₇–X₂₆₂₈₄ — Collineation mappings involving Gemini triangle 44

  contributed by Clark Kimberling, October 31, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 44, as in centers X₂₆₂₂₇-X₂₆₂₈₄. Then

  m(X) = a^2 x - (a^2 + c^2) y - (a^2 + b^2) z : : .

  . A point X lies on the Euler line if and only if m(X) also lies on the Euler line. Also, X lies on the circumcircle if and only if m(X) lies on the circumcircle; specifically, the line XX₂ meets the circumcircle in X and m(X). Moreover, m(m(X)) = X for every point X.

• X₂₆₂₉₀–X₂₆₅₂₅ — Endo-homothetic centers

  contributed by César Eliud Lozada, October 31, 2018. This section comprises the endo-homothetic centers of the family of triangles homothetic with the reference triangle ABC. This family is composed by the following 40 triangles:

  ABC, ABC-X3 reflections, anti-Aquila, anti-Ara, 5th anti-Brocard, 2nd anti-circumperp-tangential, anti-Euler, anti-inner-Grebe, anti-outer-Grebe, anti-Mandart-incircle, anticomplementary, Aquila, Ara, 1st Auriga, 2nd Auriga, 5th Brocard, 2nd circumperp tangential, Ehrmann-mid, Euler, outer-Garcia, Gossard, inner-Grebe, outer-Grebe, Johnson, inner-Johnson, outer-Johnson, 1st Johnson-Yff, 2nd Johnson-Yff, Lucas homothetic, Lucas(-1) homothetic, Mandart-incircle, medial, 5th mixtilinear, 3rd tri-squares-central, 4th tri-squares-central, X3-ABC reflections, inner-Yff, outer-Yff, inner-Yff tangents, outer-Yff tangents.

  For definitions and coordinates of these triangles, see the index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html).

• X₂₆₅₂₆–X₂₆₅₇₄ — Collineation mappings involving Gemini triangle 45

  contributed by Clark Kimberling, November 1, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 45, as in centers X₂₆₅₂₆-X₂₆₅₇₄. Then

  m(X) = (b + c - a)(b - c)^2 x + b^2 (a - b + c) y + c^2 (a + b - c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₆₅₇₅–X₂₆₆₁₂ — Collineation mappings involving Gemini triangle 46

  contributed by Clark Kimberling, November 1, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 46, as in centers X₂₆₅₇₅-X₂₆₆₁₂. Then

  m(X) = (a + b - c) (a - b + c) (b + c)^2 x + b^2 (b + c - a) (a + b - c) y + c^2 (b + c - a) (a - b + c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₆₆₂₁–X₂₆₆₅₂ — Collineation mappings involving Gemini triangle 47

  contributed by Clark Kimberling, November 2, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 47, as in centers X₂₆₆₂₁-X₂₆₆₅₂. Then

  m(X) = a^2 (a - b + c) (a + b - c) x + (b + c - a) (a + b - c) (a + c)^2 y + (b + c - a) (a - b + c) (a + b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₆₆₅₃–X₂₆₆₉₉ — Collineation mappings involving Gemini triangle 48

  contributed by Clark Kimberling, November 2, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 48, as in centers X₂₆₆₅₃-X₂₆₆₉₉. Then

  m(X) = a^2 (b + c - a) x + (a - b + c) (a - c)^2 y + (a + b - c) (a - b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₆₇₀₀–X₂₆₇₁₇ — Circumcircle-X-antipodes

  contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas and centers), November 2, 2018. Let C(P) be the circumconic with perspector P =\(p : q : r\) (barycentrics), and let U =\(u : v : w\) and F =\(f : g : h\) be distinct points, with U on C(P). Let U* be the point, other than U, that lies on C(P) and on the line FU. Then

  U* = u^2 q r (h v p + f w q + f v r) (g w p + f w q + f v r) : :

  If P = X₆, then C(P) is the circumcircle; in this case, the point U* is here named the circumcircle-F-antipode of U, given by

  U* = b^2 c^2 u^2 (a^2 h v + b^2 f w + c^2 f v)(a^2 g w + b^2 f w + c^2 f v) : :

  Note that the circumcircle-X₃-antipode of U is the ordinary antipode of U.

  This preabmle continues with lists of examples, bypified by these:

  Circumcircle-X₁-antipodes: {74, 26700}, {99, 741}, {100, 106}, {101, 105}, {102, 108}, {103, 934}, {104, 109}, {107, 26701}{110, 759}, {111, 8691}, {112, 26702}, …

  Circumcircle-X₂-antipodes: {74, 1302}, {98, 110}, {99, 111}, {100, 105}, {101, 675}, {102, 9056}, {103, 9057}, {104, 9058}, {106, 9059}, {107, 1297}, {108, 26703}, {109, 1311}, {112, 2373}, {476, 842}, {477, 9060}, …

  Circumcircle-X₃-antipodes: {74, 110}, {98, 99}, {100, 104}, {101, 103}, {102, 109}, {105, 1292}, {106, 1293}, {107, 1294}, {108, 1295}, {111, 1296}, {112, 1297}, {476, 477}, …

  Circumcircle-X₄-antipodes: {74, 107}, {98, 112}, {99, 3563}, {100, 915}, {101, 917}, {102, 26704}, {103, 26705}, {104, 108}, {105, 26706}, {110, 1300}, {477, 1304}, …

• X₂₆₇₁₈–X₂₆₇₅₁ — Centers associated with the Gemini triangles 1-10

  contributed by Randy Hutson, November 2, 2018. These Gemini triangles are introduced in the preamble just before X₂₄₅₃₇.

• X₂₆₇₅₂–X₂₆₈₀₀ — Collineation mappings involving Gemini triangle 49

  contributed by Clark Kimberling, November 3, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 49, as in centers X₂₆₇₅₂-X₂₆₈₀₂. Then

  m(X) = a (b + c)^2 x + b (a - c)^2 y + c (a - b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₆₈₀₁–X₂₆₈₆₂ — Collineation mappings involving Gemini triangle 50

  contributed by Clark Kimberling, November 3, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 50, as in centers X₂₆₈₀₁-X₂₆₈₆₂. Then

  m(X) = a (b - c)^2 x + b (a + c)^2 y + c (a + b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₆₈₆₄–X₂₆₉₅₈ — Endo-homothetic centers

  contributed by César Eliud Lozada, November 3, 2018. This section comprises the endo-homothetic centers of the family of triangles homothetic with the excentral triangle of a reference triangle ABC. This family is composed by the following 31 triangles: Ascella, Atik, 1st circumperp, 2nd circumperp, inner-Conway, Conway, 2nd Conway, 3rd Conway, 3rd Euler, 4th Euler, excenters-reflections, excentral, 2nd extouch, hexyl, Honsberger, inner-Hutson, Hutson intouch, outer-Hutson, incircle-circles, intouch, inverse-in-incircle, 6th mixtilinear, 2nd Pamfilos-Zhou, 1st Sharygin, tangential-midarc, 2nd tangential-midarc, Ursa major, Ursa minor, Wasat, Yff central, 2nd Zaniah. For definitions and coordinates of these triangles, see the index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html).

• X₂₆₉₅₉–X₂₇₀₁₉ — Collineation mappings involving Gemini triangle 51

  contributed by Clark Kimberling, November 4, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 51, as in centers X₂₆₅₉₅-X₂₇₀₁₉. Then

  m(X) = a (b - c)^2 x + b (a^2 + c^2) y + c (a^2 + b^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₀₂₀–X₂₇₀₈₁ — Collineation mappings involving Gemini triangle 52

  contributed by Clark Kimberling, November 4, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 52, as in centers X₂₇₀₂₀-X₂₇₀₈₁. Then

  m(X) = a (b + c)^2 x + b (a^2 + c^2) y + c (a^2 + b^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₀₉₁–X₂₇₁₄₁ — Collineation mappings involving Gemini triangle 53

  contributed by Clark Kimberling, November 5, 2018.Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 53, as in centers X₂₇₀₉₁-X₂₇₁₄₁. Then

  m(X) = a (b^2 + c^2) x + b (a - c)^2 y + c (a - b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₁₄₂–X₂₇₁₉₅ — Collineation mappings involving Gemini triangle 54

  contributed by Clark Kimberling, November 5, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 54, as in centers X₂₇₁₄₂-X₂₇₁₉₅. Then

  m(X) = a (b^2 + c^2) x + b (a + c)^2 y + c (a + b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₁₉₈–X₂₇₂₀₈ — Collineation mappings involving Gemini triangle 55

  contributed by Clark Kimberling, November 6, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 55, as in centers X₂₇₁₉₈-X₂₇₂₀₈. Then

  m(X) = a^2 (b^2 - 2 a c) (c^2 - 2 a b) x + 2 a c (a^2 - 2 b c) (c^2 - 2 a b) y + 2 a b (a^2 - 2 b c) (b^2 - 2 a c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₂₁₀–X₂₇₂₂₀ — Collineation mappings involving Gemini triangle 56

  contributed by Clark Kimberling, November 6, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 56, as in centers X₂₇₂₁₀-X₂₇₂₂₈. Then

  m(X) = a^2 (b^2 + 2 a c) (c^2 + 2 a b) x - 2 a c (a^2 + 2 b c) (c^2 + 2 a b) y - 2 a b (a^2 + 2 b c) (b^2 + 2 a c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₂₂₁–X₂₇₂₃₂ — Collineation mappings involving Gemini triangle 57

  contributed by Clark Kimberling, November 6, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 57, as in centers X₂₇₂₂₁-X₂₇₂₃₂. Then

  m(X) = (b^2 +c^2) x / (b^2 - b c + c^ 2) + a c y / (c^2 - c a + a^2) + a b z / (a^2 - a b + b^2) : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₂₃₂–X₂₇₂₄₅ — Collineation mappings involving Gemini triangle 58

  contributed by Clark Kimberling, November 6, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 58, as in centers X₂₇₂₃₃-X₂₇₂₄₅. Then

  m(X) = (b^2 +c^2) x / (b^2 + b c + c^ 2) - a c y / (c^2 + c a + a^2) - a b z / (a^2 + a b + b^2) : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₂₄₈–X₂₇₂₉₇ — Collineation mappings involving Gemini triangle 59

  contributed by Clark Kimberling, November 7, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 59, as in centers X₂₇₂₄₈-X₂₇₂₉₇. Then

  m(X) = a(a b + a c - b c) x + b (a b + a c + b c) y + c (a b + a c + b c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₂₉₉–X₂₇₃₅₁ — Collineation mappings involving Gemini triangle 60

  contributed by Clark Kimberling, November 7, 2018. Extending the preambles just before X₂₄₅₃₇ and X₂₆₁₅₃, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 60, as in centers X₂₇₂₉₉-X₂₇₃₅₁. Then

  m(X) = a(a b + a c + b c) x + b (a b - a c + b c) y + c (a c - a b + b c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₃₇₈–X₂₇₄₂₂ — Collineation mappings involving Gemini triangle 61

  contributed by Clark Kimberling, November 8, 2018. Following is a list of central triangles, by barycentric coordinates of A-vertex. The full names are Gemini triangle 61, Gemini triangle 62, etc. See the preamble just before X₂₄₅₃₇ and X₂₆₁₅₃ for definitions of Gemini triangles 1-60. (Clark Kimberling, November 8, 2018)

  Gemini 61 a (a + b + c) (a - b - c) : (b + c) (a - b + c) (a + b - c) : (b + c) (a - b + c)(a + b - c)

  Let A″B″C″ = Gemini triangle 1, defined by A' = a : b + c : b + c, and let A'B'C' = Gemini triangle 61.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 62 (b c + a c - a b ) (a b - a c + b c) (b^2 + b c + c^2) : (a b - a c - b c) (a b + a c - b c)(c^2 + a b) : (a c - a b - b c) (a c + a b - b c)(b^2 + a c)

  Let A″B″C″ = Gemini triangle 3, defined by A' = a : a + b : a + c, and let A'B'C' = Gemini triangle 62.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 63 - (b^2 + b c + c^2) : a b + 2 a c + 2 b c + c^2 : a c + 2 a b + 2 b c + b^2

  Let A″B″C″ = Gemini triangle 4, defined by A' = -a : a + b : a + c, and let A'B'C' = Gemini triangle 63.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 64 (a b - a c + b c) (a b - a c - b c) (b^2 + c^2 - b c) : (-a b + a c + b c) (a b + a c - b c)(c^2 + a b - 2 b c) : (a b - a c + b c) (a b + a c - b c) (b^2 + a c - 2 b c)

  Let A″B″C″ = Gemini triangle 5, defined by A' = -a : a - b : a - c, and let A'B'C' = Gemini triangle 64.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 65 (a - b - c) (a + b + c) : (a + b - c) (a - b + c) : (a + b - c) (a - b + c)

  Let A″B″C″ = Gemini triangle 13, defined by A' = b +\(c : a : a\), and let A'B'C' = Gemini triangle 65.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 66 (b + c)(a^2 + b^2 + c^2 + 2 a b + 2 a c + b c) : -a (a + b) (a + c) : -a (a + b) (a + c)

  Let A″B″C″ = Gemini triangle 21, defined by A' = a + b +\(c : a : a\), and let A'B'C' = Gemini triangle 66.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 67 (2 a + b + c) (a^2 + b^2 + c^2 + 2 a b + 2 a c + b c) : a (a + 2 b + c) (a + b + 2 c) : a (a + 2 b + c)(a + b + 2 c)

  Let A″B″C″ = Gemini triangle 22, defined by A' = a + b +\(c : -a : -a\), and let A'B'C' = Gemini triangle 67.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 68 a (b^2 + c^2 + a b + c a + b c) : -b c (b + c) : -b c (b + c)

  Let A″B″C″ = Gemini triangle 23, defined by A' = a + b + c : b + c : b + c, and let A'B'C' = Gemini triangle 68.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 69 (b + c) (b^2 + c^2 - a^2 - b c) : -a (a + b) (a + c) : -a (a + b) (a + c)

  Let A″B″C″ = Gemini triangle 27, defined by A' = a - b \(- c : a : a\), and let A'B'C' = Gemini triangle 69.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 70 (a - 2 b -2 c) (b^2 + c^2 + a b + a c - b c) : (b + c) (2 a + 2 b - c) (2 a - b + 2 c) : (b + c) (2 a + 2 b - c) (2 a - b + 2 c)

  Let A″B″C″ = Gemini triangle 28, defined by A' = a - b - c : b + c : b + c, and let A'B'C' = Gemini triangle 70.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 71 a ( a - 3 b + c) (a + b - 3 c) : (3 a - b - c) (3 c - a - b) (b - c) : (3 a - b - c) (a - 3 b + c) (b - c)

  Let A″B″C″ = Gemini triangle 30, defined by A' = a : c - b : b - c, and let A'B'C' = Gemini triangle 71.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 72 b c (a^2 - b c)^2 : a^2 (b^2 - a c) (c^2 - a b) : a^2 (b^2 - a c) (c^2 - a b)

  Let A″B″C″ = Gemini triangle 31, defined by A' = b c : a^2 : a^2, and let A'B'C' = Gemini triangle 72.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 73 b c (a^4 - b^2 c^2) : a^2 (b^2 + a c) (c^2 + a b) : a^2 (b^2 + a c) (c^2 + a b)

  Let A″B″C″ = Gemini triangle 32, defined by A' = -b c : a^2 : a^2, and let A'B'C' = Gemini triangle 73.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 74 (a^2 - b c)^2 : (b^2 - a c) (c^2 - a b) : (b^2 - a c) (c^2 - a b)

  Let A″B″C″ = Gemini triangle 33, defined by A' = a^2 : b c : b c, and let A'B'C' = Gemini triangle 74.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 75 a^4 - b^2 c^2 : -(b^2 + a c) (c^2 + a b) : -(b^2 + a c) (c^2 + a b)

  Let A″B″C″ = Gemini triangle 34, defined by A' = -a^2 : b c : b c, and let A'B'C' = Gemini triangle 75.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 76 (a + b + c) (a^2 + b^2 + c^2 - 2 b c) : 2 b c (a - b - c) : 2 b c (a - b - c)

  Let A″B″C″ = Gemini triangle 35, defined by A' = cos \(A : 1 : 1\) = -a^2 + b^2 + c^2 : 2 b c : 2 b c, and let A'B'C' = Gemini triangle 76.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 77 (a - b - c)^2 (a^2 + b^2 + c^2 - 2 b c) : -2 b c (a - b + c) (a + b - c) : - 2 b c (a - b + c) (a + b - c)

  Let A″B″C″ = Gemini triangle 36, defined by A' = -cos \(A : 1 : 1\) = a^2 - b^2 - c^2 : 2 b c : 2 b c, and let A'B'C' = Gemini triangle 77.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 78 (a + b + c) (a^2 + b^2 + c^2 - 2 b c) : (b + c - a) (a^2 - b^2 - c^2) : (b + c - a) (a^2 - b^2 - c^2)

  Let A″B″C″ = Gemini triangle 37, defined by A' = sec \(A : 1 : 1\) = 2 b c : -a^2 + b^2 + c^2 : -a^2 + b^2 + c^2, and let A'B'C' = Gemini triangle 78.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 79 (a - b - c)^2 (a^2 + b^2 + c^2 - 2 b c) : (a - b + c) (a + b - c) (b^2 + c^2 - a^2) : (a - b + c) (a + b - c) (b^2 + c^2 - a^2)

  Let A″B″C″ = Gemini triangle 38, defined by A' = -sec \(A : 1 : 1\) = -2 b c : -a^2 + b^2 + c^2 : -a^2 + b^2 + c^2, and let A'B'C' = Gemini triangle 79.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 80 a (b^2 + c^2 + a b + a c) : -b c (a + b + c) : -b c (a + b + c)

  Let A″B″C″ = Gemini triangle 39, defined by A' = -a + b + c : a + b + c : a + b + c, and let A'B'C' = Gemini triangle 80.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 81 a (b^2 + c^2 + a b + a c) : b c (a - b - c) : b c (a - b - c)

  Let A″B″C″ = Gemini triangle 40, defined by A' = a + b + c : -a + b + c : -a + b + c, and let A'B'C' = Gemini triangle 81.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 82 (a^2 - b^2 - c^2) (a^2 + b^2 + c^2 : (a^2 + b^2 - c^2) (a^2 - b^2 + c^2 : (a^2 + b^2 - c^2) (a^2 - b^2 + c^2

  Let A″B″C″ = Gemini triangle 41, defined by A' = b^2 + c^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 82.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 83 (b^2 + c^2) (a^2 + b^2 + c^2 - b c) (a^2 + b^2 + c^2 + b c) : - a^2 (a^2 + b^2) (a^2 + c^2) : - a^2 (a^2 + b^2) (a^2 + c^2)

  Let A″B″C″ = Gemini triangle 42, defined by A' = a^2 + b^2 + c^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 83.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 84 a^2 (a^2 + b^2 + c^2) (a^2 - b^2 - c^2) : (b^2 + c^2) (a^2 - b^2 + c^2) (a^2 + b^2 - c^2) : (b^2 + c^2)(a^2 - b^2 + c^2) (a^2 + b^2 - c^2)

  Let A″B″C″ = Gemini triangle 43, defined by A' = a^2 : b^2 + c^2 : b^2 + c^2 , and let A'B'C' = Gemini triangle 84.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 85 a^2 - a b - a c + 2 b c : a (a - b - c) : a (a - b - c)

  Let A″B″C″ = Gemini triangle 45, defined by A' = (b - c)^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 85.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 86 (a - b - c) (a^2 + a b + a c + 2 b c) : a (a - b + c) (a + b - c) : a (a - b + c) (a + b - c)

  Let A″B″C″ = Gemini triangle 46, defined by A' = (b + c)^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 86.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 87 a (a - b - c) (a^2 + a b + a c + 2 b c) : (b + c)^2 (a - b + c) (a + b - c) : (b + c)^2 (a - b + c)(a + b - c)

  Let A″B″C″ = Gemini triangle 47, defined by A' = a^2 : (b + c)^2 : (b + c)^2 , and let A'B'C' = Gemini triangle 87.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 88 a (a^2 - a b - a c + 2 b c) : (b - c)^2 (a - b - c) : (b - c)^2 (a - b - c)

  Let A″B″C″ = Gemini triangle 48, defined by A' = a^2 : (b - c)^2 : (b - c)^2, and let A'B'C' = Gemini triangle 88.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 89 (b + c)(a^2 + b c) : -a (b - c)^2 : -a (b - c)^2

  Let A″B″C″ = Gemini triangle 49, defined by A' = (b + c)^2 : (b - c)^2 : (b - c)^2, and let A'B'C' = Gemini triangle 89.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 90 a^2 + b c : -a b - a c : -a b - a c

  Let A″B″C″ = Gemini triangle 50, defined by A' = (b - c)^2 : (b + c)^2 : (b + c)^2, and let A'B'C' = Gemini triangle 90.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 91 a^2 b+a^2 c - 2 a b c + b^2 c + b c^2 : -a (b^2 + c^2) : -a (b^2 + c^2)

  Let A″B″C″ = Gemini triangle 51, defined by A' = (b - c)^2 : b^2 + c^2 : b^2 + c^2, and let A'B'C' = Gemini triangle 91.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 92 a^2 b + a^2 c + 2 a b c + b^2 c + b c^2 : -a (b^2 + c^2) : -a (b^2 +c ^2)

  Let A″B″C″ = Gemini triangle 52, defined by A' = (b + c)^2 : b^2 + c^2 : b^2 + c^2, and let A'B'C' = Gemini triangle 92.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 93 a^2 b + a^2 c - 2 a b c + b^2 c + b c^2 : -a (b - c)^2 : -a (b - c)^2

  Let A″B″C″ = Gemini triangle 53, defined by A' = b^2 + c^2 : (b - c)^2 : (b - c)^2, and let A'B'C' = Gemini triangle 93.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 94 a^2 b + a^2 c + 2 a b c + b^2 c + b c^2 : -a (b + c)^2 : -a (b + c)^2

  Let A″B″C″ = Gemini triangle 54, defined by A' = b^2 + c^2 : (b + c)^2 : (b + c)^2 , and let A'B'C' = Gemini triangle 94.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 95 (a^2 - 2 b c) (4 a^2 - b c) : 2 (b^2 - 2 a c) (c^2 - 2 a b) : 2 (b^2 - 2 a c)(c^2 - 2 a b)

  Let A″B″C″ = Gemini triangle 55, defined by A' = a^2 : 2 b c : 2 b c, and let A'B'C' = Gemini triangle 95.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 96 (a^2 + 2 b c) (4 a^2 - b c) : -2 (b^2 + 2 a c) (c^2 + 2 a b) : -2 (b^2 + 2 a c) (c^2 + 2 a b)

  Let A″B″C″ = Gemini triangle 56, defined by A' = -a^\(2 : 2bc : 2bc\) , and let A'B'C' = Gemini triangle 96.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 97 (b^2 - b c + c^2) (a^4 + a^2 b^2 + a^2 c^2 - a^2 b c + b^2 c^2) : -b c (a^2 - a b + b^2) (a^2 - a c + c^2) : -b c (a^2 - a b + b^2) (a^2 - a c + c^2)

  Let A″B″C″ = Gemini triangle 57, defined by A' = b^2 + c^2 : b c : b c, and let A'B'C' = Gemini triangle 97.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 98 (b^2 + b c + c^2) (a^4 + a^2 b^2 + a^2 c^2 - a^2 b c + b^2 c^2) : b c (a^2 + a b + b^2) (a^2 + a c + c^2) : b c (a^2 + a b + b^2) (a^2 + a c + c^2)

  Let A″B″C″ = Gemini triangle 58, defined by A' = b^2 + c^2 : -b c : -b c, and let A'B'C' = Gemini triangle 98.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 99 a b^2 + a c^2 + b^2 c + b c^2 : -a (b c + c a + a b): -a (b c + c a + a b)

  Let A″B″C″ = Gemini triangle 59, defined by A' = -b c + c a + a b : b c + c a + a b : b c + c a + a b, and let A'B'C' = Gemini triangle 99.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).
  Gemini 100 ab^2+ac^2+b^2 c+bc^2 : a (b c - c a - a b): a (b c - c a - a b)

  Let A″B″C″ = Gemini triangle 60, defined by A' = -b c + c a + a b : b c + c a + a b : b c + c a + a b, and let A'B'C' = Gemini triangle 100.

  The collineation (A, B, C,X₂; A',B',C',X₂) is the inverse of the collineation (A, B, C,X₂; A″,B″,C″,X₂).

  If T is a central triangle A'B'C' with A' of the form \(f(a,b,c) : g(a,b,c) : g(a,b,c)\), then the (A, B, C,X₂; A',B',C',X₂) collineation image of the Euler line is the Euler line. Examples include Gemini triangles 61, 65-70, and 72-100.

• X₂₇₄₂₄–X₂₇₄₇₀ — Collineation mappings involving Gemini triangle 62

  contributed by Clark Kimberling, November 8, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 62, as in centers X₂₇₄₂₄-X₂₇₄₇₀. Then

  m(X) = (bc+ac-ab)(ab-ac+bc)(b^2+bc+c^2)x + (c^2+ab)(ab-ac+bc)ab-ac-bc)y + (b^2+ac)(ac-ab-bc)(ac-ab-bc)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₄₇₁–X₂₇₄₉₅ — Collineation mappings involving Gemini triangle 63

  contributed by Clark Kimberling, November 9, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 63, as in centers X₂₇₄₇₁-X₂₇₄₉₅. Then

  m(X) = -(b^2+bc+c^2)x + (ab+2ac+2bc+c^2)y + (ac+2ab+2bc+b^2)z : : .

• X₂₇₄₉₆–X₂₇₅₀₃ — Collineation mappings involving Gemini triangle 64

  contributed by Clark Kimberling, November 9, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 64, as in centers X₂₇₄₉₆-X₂₇₅₀₃. Then

  m(X) = (ab-ac-bc)(ab-ac+bc)(b^2-bc+c^2)x + (ab-ac-bc)(ab-ac+bc)(ab-2ac+c^2)y : (ac-ab-bc)(ac-ab+bc)(ac-2ab+b^2)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₅₀₄–X₂₇₅₄₉ — Collineation mappings involving Gemini triangle 65

  contributed by Clark Kimberling, November 9, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 65, as in centers X₂₇₅₀₄-X₂₇₅₄₉. Then

  m(X) = a(a+b+c)(a-b-c)x - b(a+b-c)(a-b-c) - c(a-b+c)(a-b-c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 9, 2018)

• X₂₇₅₅₃–X₂₇₅₈₉ — Collineation mappings involving Gemini triangle 66

  contributed by Clark Kimberling, November 10, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 66, as in centers X₂₇₅₅₃-X₂₇₅₈₉. Then

  m(X) = (b+c)(a^2+b^2+c^2+2ab+2ac+bc)x - a(a+b)(a+c)y - a(a+b)(a+c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₅₉₀–X₂₇₆₂₀ — Collineation mappings involving Gemini triangle 67

  contributed by Clark Kimberling, November 11, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 67, as in centers X₂₇₅₉₀-X₂₇₆₂₀. Then

  m(X) = (2a+b+c)(a^2+b^2+c^2+2ab+2ac+bc)x + b(2a+b+c)(a+b+2c)y + c(2a+b+c)(a+2b+c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₆₂₁–X₂₇₆₈₂ — Collineation mappings involving Gemini triangle 68

  contributed by Clark Kimberling, November 11, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 68, as in centers X₂₇₆₂₁-X₂₇₆₈₂. Then

  m(X) = a(b^2+c^2+ab+ca+bc)x - ac(a+c)y - ab(a+b)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line. Fixed points of m include X₂, X₃₆, X₂₃₈, and X₆₆₇.

• X₂₇₆₈₅–X₂₇₇₃₅ — Collineation mappings involving Gemini triangle 69

  contributed by Clark Kimberling, November 12, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 69, as in centers X₂₇₆₈₅-X₂₇₇₃₅. Then

  m(X) = (b+c)(b^2+c^2-a^2-bc)x - b(a+b)(b+c)y - c(a+c)(b+c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₇₃₆–X₂₇₇₇₇ — Collineation mappings involving Gemini triangle 70

  contributed by Clark Kimberling, November 12, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 70, as in centers X₂₇₇₃₆-X₂₇₇₇₇. Then

  m(X) = (b+c)(b^2+c^2-a^2-bc)x - b(a+b)(b+c)y - c(a+c)(b+c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₇₈₂–X₂₇₈₁₂ — Centers associated with Gemini triangles 11-18

  contributed by Randy Hutson, November 12, 2018. Gemini triangles are introduced in the preamble just before X₂₄₅₃₇.

• X₂₇₈₁₃–X₂₇₈₃₇ — Collineation mappings involving Gemini triangle 71

  contributed by Clark Kimberling, November 13, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 71, as in centers X₂₇₈₁₃-X₂₇₈₃₇. Then

  m(X) = a(a-3b+c)(a+b-3c)x + (a-c)(a-3b+c)(a+b-3c)y + (a-b)(a-3b+c)(a+b-3c)z : : .

• X₂₇₈₃₈–X₂₇₈₆₅ — Collineation mappings involving Gemini triangle 72

  contributed by Clark Kimberling, November 13, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 72, as in centers X₂₇₈₃₈-X₂₇₈₆₅. Then

  m(X) = bc(a^2-bc)^2x + b^2(a^-bc)(c^2-ab)y + c^2(a^2-bc)(b^2-ac)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₈₇₂–X₂₇₉₀₆ — Collineation mappings involving Gemini triangle 73

  contributed by Clark Kimberling, November 14, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 73, as in centers X₂₇₈₇₂-X₂₇₉₀₆. Then

  m(X) = bc(a^2-bc)^2x + b^2(a^2-bc)(c^2-ab)y + c^2(a^2-bc)(b^2-ac)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₉₀₇–X₂₇₉₅₂ — Collineation mappings involving Gemini triangle 74

  contributed by Clark Kimberling, November 14, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 74, as in centers X₂₇₉₀₇-X₂₇₉₅₂. Then

  m(X) = bc(a^2-bc)^2x + ac(a^2-bc)(c^2-ab)y + ab(a^2-bc)(b^2-ac)z : :

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₇₉₅₄–X₂₈₀₁₀ — Collineation mappings involving Gemini triangle 75

  contributed by Clark Kimberling, November 15, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 75, as in centers X₂₇₉₅₄-X₂₈₀₁₀. Then

  m(X) = bc(a^4-b^2c^2)x - ac(a^2+bc)(c^2+ab)y - ab(a^2+bc)(b^2+ac)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

Part 15

• X₂₈₀₁₁–X₂₈₀₄₂ — Collineation mappings involving Gemini triangle 76

  contributed by Clark Kimberling, November 15, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 76, as in centers X₂₈₀₁₁-X₂₈₀₄₂. Then

  m(X) = bc(a^4-b^2c^2)x - ac(a^2+bc)(c^2+ab)y - ab(a^2+bc)(b^2+ac)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₀₄₃–X₂₈₀₇₃ — Collineation mappings involving Gemini triangle 77

  contributed by Clark Kimberling, November 16, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 77, as in centers X₂₈₀₄₃-X₂₈₀₇₃. Then

  m(X) = (b+c-a)^2(a^2+b^2+c^2-2bc)x - 2ac(b+c-a)(a+b-c)y - 2ab(b+c-a)(a-b+c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₀₇₄–X₂₈₁₁₇ — Collineation mappings involving Gemini triangle 78

  contributed by Clark Kimberling, November 17, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 78, as in centers X₂₈₀₇₄-X₂₈₁₁₇. Then

  m(X) = (a+b+c)(a^2+b^2+c^2-2bc)x - (a-b+c)(a^2-b^2-c^2)y + (a+b-c)(a^2-b^2-c^2)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₁₁₈–X₂₈₁₄₂ — Collineation mappings involving Gemini triangle 79

  contributed by Clark Kimberling, November 17, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 79, as in centers X₂₈₁₁₈-X₂₈₁₄₂. Then

  m(X) = (a-b-c)^2(a^2+b^2+c^2-2bc)x + (b+c-a)(a+b-c)(a^2-b^2-c^2)y + (b+c-a)(a-b+c)(a^2-b^2-c^2) : :

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₁₄₅–X₂₈₂₃₆ — Points on circumcircle and line at infinity

  contributed by Clark Kimberling, November 17, 2018. Suppose that X =\(x : y : z\) is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X″ be the circumcircle-antipode of X', and let X’’’ be its isogonal conjugate, on the line at infinity. As a direction, X’’’ is perpendicular to X. In this section, X is given by the form (b - c)(h a + k(b + c)) : : , where h and k are constants.

  In the table below, Columns 1 and 2 show h and k.

  Column 3. (b - c)(h a + k(b + c)) : : , on infinity line, referenced below as \(x : y : z\)

  Column 4. (isogonal conjugate of \(x : y : z\)) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as \(u : v : w\)

  Column 5. (antipode of \(u : v : w\)) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2(a^2-b^2-c^2)v + 2a^2(a^2-b^2-c^2)w : : on circumcircle, referenced below as \(u1 : v1 : w1\)

  Column 6. (isogonal conjugate of \(u1 : v1 : w1\)) = a^2/u1 + b^2/v1 + c^2/w1

  For each row, let X be the point in Column 3 and X' the point in Column 6. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular. (The table is omitted here.)

• X₂₈₂₃₈–X₂₈₂₉₀ — Collineation mappings involving Gemini triangle 80

  contributed by Clark Kimberling, November 18, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 80, as in centers X₂₈₂₃₈-X₂₈₂₉₀. Then

  m(X) = a(b^2+c^2+ab+ac)x - ac(a+b+c)y - ab(a+b+c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₃₄₈–X₂₈₄₀₃ — Collineation mappings involving Gemini triangle 81

  contributed by Clark Kimberling, November 19, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 81, as in centers X₂₈₃₄₈-X₂₈₄₀₃. Then

  m(X) = a(b^2+c^2+ab+ac)x + ac(b-c-a)y - ab(c-b-a)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₄₀₄–X₂₈₄₄₁ — Collineation mappings involving Gemini triangle 82

  contributed by Clark Kimberling, November 26, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 82, as in centers X₂₈₄₀₄-X₂₈₄₄₁. Then

  m(X) = a^2(a^2-b^2-c^2)(a^2+b^2+c^2)x - b^2(a^2-b^2-c^2)(a^2+b^2-c^2)y - c^2(a^2-b^2-c^2)(a^2-b^2+c^2)a : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₄₆₇–X₂₈₅₈₅ — Points on circumcircle and line at infinity

  Suppose that X =\(x : y : z\) is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X″ be the circumcircle-antipode of X', and let X’’’ be its isogonal conjugate, on the line at infinity. As a direction, X’’’ is perpendicular to X. In this section, X is given by the form h (- 2 a^3 + b^3 + c^3) + j (a^2 (b + c) + k a ( b^2 + c^2) - ( j + k) (b c^2 + b^2 c) : : , where h, j, k are constants. (Clark Kimberling, November 27, 2018)

  In the table below, Columns 1-3 show h, j, k.

  Column 4. h (- 2 a^3 + b^3 + c^3) + j (a^2 (b + c) + k a ( b^2 + c^2) - ( j + k) (b c^2 + b^2 c) : : , on infinity line, referenced below as \(x : y : z\)

  Column 5. (isogonal conjugate of \(x : y : z\)) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as \(u : v : w\)

  Column 6. (antipode of \(u : v : w\)) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2 (a^2-b^2-c^2)v + 2a^2 (a^2-b^2-c^2)w : : on circumcircle, referenced below as \(u1 : v1 : w1\)

  Column 7. (isogonal conjugate of \(u1 : v1 : w1\)) = a^2/u1 + b^2/v1 + c^2/w1

  For each row, let X be the point in Column 4 and X' the point in Column 7. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular. (The table is omitted here.).

• X₂₈₅₈₆–X₂₈₆₆₁ — Centers associated with the Gemini triangles 2-28

  contributed by Randy Hutson, November 27, 2018. The Gemini triangles are introduced in the preamble just before X₂₄₅₃₇.

• X₂₈₆₆₃–X₂₈₆₉₃ — Collineation mappings involving Gemini triangle 83

  contributed by Clark Kimberling, November 27, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 83, as in centers X₂₈₆₆₃-X₂₈₆₉₃. Then

  m(X) = (b^2 + c^2) (a^2 + b^2 + c^2 - b c) (a^2 + b^2 + c^2 + b c) x - b^2 (a^2 + b^2) (b^2 + c^2) y - c^2 (a^2 + c^2) (b^2 + c^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₆₉₄–X₂₈₇₃₃ — Collineation mappings involving Gemini triangle 84

  contributed by Clark Kimberling, November 27, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 84, as in centers X₂₈₆₉₄-X₂₈₇₃₃. Then

  m(X) = a^2(a^2 - b^2 - c^2) (a^2 + b^2 + c^2) x + (a^2 + c^2) (a^2 + b^2 - c^2) (-a^2 + b^2 + c^2) y + (a^2 + b^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₇₃₄–X₂₈₇₈₀ — Collineation mappings involving Gemini triangle 85

  contributed by Clark Kimberling, November 29, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 85, as in centers X₂₈₇₃₄-X₂₈₇₈₀. Then

  m(X) = a (a^2 - a b - a c 2 b c) x - b^2(a - b + c) y - c^2 (a + b - c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₇₈₉–X₂₈₈₃₇ — Collineation mappings involving Gemini triangle 86

  contributed by Clark Kimberling, November 30, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 86, as in centers X₂₈₇₈₉-X₂₈₈₃₇. Then

  m(X) = a (a - b - c) (a^2 + a b + a c + 2 b c) x + b^2 (b + c - a) (a + b - c) y + c^2 (b + c - a) (a - b + c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₈₃₈–X₂₈₉₁₅ — Points on circumcircle and line at infinity

  contributed by Clark Kimberling, November 26, 2018. Suppose that X =\(x : y : z\) is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X″ be the circumcircle-antipode of X', and let X’’’ be its isogonal conjugate, on the line at infinity. As a direction, X’’’ is perpendicular to X. In this section, X is given by the form (b - c) (h a^2 + j (b^2 + c^2) + k b c + (h - j + k)(a b + a c) : : , where h, j, k are constants.

  In the table below, Columns 1-3 show h, j, k.

  Column 4. (b - c) (h a^2 + j (b^2 + c^2) + k b c + (h - j + k)(a b + a c) : : , on infinity line, referenced below as \(x : y : z\)

  Column 5. (isogonal conjugate of \(x : y : z\)) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as \(u : v : w\)

  Column 6. (antipode of \(u : v : w\)) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2 (a^2-b^2-c^2)v + 2a^2 (a^2-b^2-c^2)w : : on circumcircle, referenced below as \(u1 : v1 : w1\)

  Column 7. (isogonal conjugate of \(u1 : v1 : w1\)) = a^2/u1 + b^2/v1 + c^2/w1

  For each row, let X be the point in Column 4 and X' the point in Column 7. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular.

  In the table below, the points in Column 4 are here given names of the form Point Propus(h,j,k). (The table is omitted here.)

• X₂₈₉₁₆–X₂₈₉₅₀ — Collineation mappings involving Gemini triangle 87

  contributed by Clark Kimberling, December 2, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 87, as in centers X₂₈₉₁₆-X₂₈₉₆₀. Then

  m(X) = a(b+c-a)(a^2+ab+ac+2bc)x + (a+c)^2(a+b-c)(a-b-c)y + (a+b)^2(a-b+c)(a-b-c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₈₉₆₁ — Collineation mappings involving Gemini triangle 88

  contributed by Clark Kimberling, December 1, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 88, as in centers X₂₈₉₆₁-X₂₉₀₀₇. Then

  m(X) = a(a^2-ab-ac+2bc)x - (a-c)^2(a-b+c)y + (a+b)^2(a+b-c)z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₀₀₉–X₂₉₁₅₇ — Points on circumcircle and line at infinity

  contributed by Clark Kimberling, December 3, 2018. Suppose that X =\(x : y : z\) is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X″ be the circumcircle-antipode of X', and let X’’’ be its isogonal conjugate, on the line at infinity. As a direction, X’’’ is perpendicular to X.

  In this section, X is given by the form

  (b - c) (h a^3 + i (b^3 + c^3) + j a^2 (b + c) + (i - h) (b c^2 + b^2 c) + k a b c) : : , where h, i, j, k are constants.

  In the table below, Columns 1-4 show h, i, j, k.

  Column 5. (b - c) (h a^3 + i (b^3 + c^3) + j a^2 (b + c) + (i - h) (b c^2 + b^2 c) + k a b c) : : , on infinity line, referenced below as \(x : y : z\)

  Column 6. (isogonal conjugate of \(x : y : z\)) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as \(u : v : w\)

  Column 7. (antipode of \(u : v : w\)) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2 (a^2-b^2-c^2)v + 2a^2 (a^2-b^2-c^2)w : : on circumcircle, referenced below as \(u1 : v1 : w1\)

  Column 8. (isogonal conjugate of \(u1 : v1 : w1\)) = a^2/u1 + b^2/v1 + c^2/w1

  For each row, let X be the point in Column 5 and X' the point in Column 8. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular.

  In the table below, the points in Column 5 are here given names of the form Point Polaris(h,i,j,k). (The table is omitted here.)

• X₂₉₃₇₅–X₂₉₄₃₂ — Collineation mappings involving Gemini triangle 89

  contributed by Clark Kimberling, December 6, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 89, as in centers X₂₈₃₇₅-X₂₉₄₃₂. Then

  m(X) = (b + c) (a^2 + b c) x + b(c - a) y - c (b - a) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₄₃₃–X₂₉₄₉₄ — Collineation mappings involving Gemini triangle 90

  contributed by Clark Kimberling, December 7, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 90, as in centers X₂₈₄₃₃-X₂₉₄₉₄. Then

  m(X) = (b + c) (a^2 + b c) x - b (a + c)^2 y - c (a + b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₄₉₇–X₂₉₅₅₁ — Collineation mappings involving Gemini triangle 91

  contributed by Clark Kimberling, December 8, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 91, as in centers X₂₈₄₉₇-X₂₉₅₅₁. Then

  m(X) = (a^2 b + a^2 c - 2 a b c + b^2 c + b c^2) x -b (a^2 + c^2) y - c (a^2 + b^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₅₅₂–X₂₉₅₆₈ — Collineation mappings involving Gemini triangle 92

  contributed by Clark Kimberling, December 8, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 92, as in centers X₂₈₅₅₂-X₂₉₅₆₈. Then

  m(X) = (a^2 b + a^2 c + 2 a b c + b^2 c + b c^2) x - b (a^2 + c^2) y - c (a^2 + b^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₅₆₉–X₂₉₆₃₀ — Points Capella(h,j,k)

  contributed by Clark Kimberling, December 8, 2018. Definition: Point Capella(h,j,k) =\(f(h,j,k,a,b,c) : f(h,j,k,b,c,a) : f(h,j,k,c,a,b)\) (barycentrics), where

  \(f(h,j,k,a,b,c) = h a^{2} + j (b^{2} + c^{2}) + k b c + (h - j + k)(a b + a c)\),

  where h, j, k are real numbers, not all zero. These points lie on the line X₁X₂.

• X₂₉₆₃₁–X₂₉₆₉₀ — Points Castor(h,j,k,p,q)

  contributed by Clark Kimberling, December 9, 2018. Definition: Point Castor(h,j,k,p,q,a,b,c) =\(f(h,j,k,p,q,a,b,c) : f(h,j,k,p,q,b,c,a) : f(h,j,k,p,q,c,a,b)\) (barycentrics), where

  \(f(h,j,k,p,q,a,b,c) = h (a^{3} + b^{3} + c^{3}) + j (a^{2} b + b^{2} c + c^{2} a + a^{2} c + b^{2} a + c^{2} b) + k (a b c) + a (p (a^{2} + b^{2} + c^{2}) + q (b c + c a + a b))\),

  where h, j, k, p, q are real numbers, not all zero. These points lie on the line X₁X₂.

• X₂₉₆₉₁–X₂₉₇₄₁ — Collineation mappings involving Gemini triangle 93

  contributed by Clark Kimberling, December 9, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 93, as in centers X₂₈₆₉₁-X₂₉₇₄₁. Then

  m(X) = (a^2 b + a^2 c - 2 a b c + b^2 c + b c^2) x - b (a - c)^2) y - c (a - b)^2 z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₇₄₂–X₂₉₈₁₃ — Collineation mappings involving Gemini triangle 94

  contributed by Clark Kimberling, December 10, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 94, as in centers X₂₉₇₄₂-X₂₉₈₁₃. Then

  m(X) = (a^2 b + a^2 c + 2 a b c + b^2 c + b c^2) x - b (a + c)^2 y - c (a + b^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₈₇₅–X₂₉₈₉₄ — Collineation mappings involving Gemini triangle 95

  contributed by Clark Kimberling, December 11, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 95, as in centers X₂₉₈₇₅-X₂₉₈₉₄. Then

  m(X) = b c (a^2 - 2 b c)(4 a^2 - b c) x + 2 a c (a^2 - 2 b c)(c^2 - 2 a b) y + 2 a b (a^2 - 2 b c)(c^2 - 2 a b) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₈₉₅–X₂₉₉₁₇ — Collineation mappings involving Gemini triangle 96

  contributed by Clark Kimberling, December 12, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 96, as in centers X₂₉₈₉₅-X₂₉₉₁₇. Then

  m(X) = b c (4 a^2 - b c) (a^2 + 2 b c) (a^2 + 2 b c) x -2 a c(a^2 + 2 b c) (c^2 + 2 a b) y - 2 a b (a^2 + 2 b c)( b^2 + 2 a c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₉₁₈–X₂₉₉₃₅ — Collineation mappings involving Gemini triangle 97

  contributed by Clark Kimberling, December 12, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 97, as in centers X₂₉₉₁₈-X₂₉₉₃₅. Then

  m(X) = (b + c) (b^2 + c^2 - a^2 - b c) x - b (a + b) (b + c) y - c (a + c) (b + c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₉₃₆–X₂₉₉₅₆ — Collineation mappings involving Gemini triangle 98

  contributed by Clark Kimberling, December 12, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 98, as in centers X₂₉₉₃₆-X₂₉₉₅₆. Then

  m(X) = (b^2 + b c + c^2) (a^4 + a^2 b^2 + a^2 c^2 + b^2 c^2 - a^2 b c) x + a c (a^2 + a b + b^2) (b^2 + b c + c^2) y + a b (a^2 + a c + c^2) (b^2 + b c + c^2) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₂₉₉₆₀–X₃₀₀₂₆ — Collineation mappings involving Gemini triangle 99

  contributed by Clark Kimberling, December 13, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 99, as in centers X₂₉₉₆₀-X₃₀₀₂₆. Then

  m(X) = (a b^2 + a c^2 + b^2 c + b c^2) x - b (b c + c a + a b) y - c (b c + c a + a b) z : : ,

  and m(X) is on the Euler line.

Part 16

• X₃₀₀₃₀–X₃₀₀₉₉ — Collineation mappings involving Gemini triangle 100

  contributed by Clark Kimberling, December 12, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 100, as in centers X₃₀₀₃₀-X₃₀₀₉₉. Then

  m(X) = (a b^2 + a c^2 + b^2 c + b c^2) x - b (b c + a b - a c) y - c (b c - a b + a c) z : : ,

  and m(X) is on the Euler line if and only if X is on the Euler line.

• X₃₀₁₀₃–X₃₀₁₇₈ — Points Celaeno(h,i,j,k,u,v,w)

  contributed by Clark Kimberling, December 14, 2018. Point Celaeno(h,i,j,k,u,v,w,a,b,c) =\(f(h,i,j,k,u,v,w,a,b,c) : f(h,i,j,k,u,v,w,b,c,a) : f(h,i,j,k,u,v,w,c,a,b)\) (barycentrics), where

  \(f(h,i,j,k,u,v,w,a,b,c) = h (a^{4} + b^{4} + c^{4}) + i (a^{3} b + b^{3} c + c^{3} a + a^{3} c + b^{3} a + c^{3} b) + j (b^{2} c^{2} + c^{2} a^{2} + a^{2} b^{2}) + k (a^{2} b c + a b^{2} c + a b c^{2}) + a (u (a^{3} + b^{3} + c^{3}) + v (a^{2} b + b^{2} c + c^{2} a + a^{2} c + b^{2} a + c^{2} b) + w a b c)\)

  where h, i, j, k, u, v, w are real numbers, not all zero. These points lie on the line X₁X₂.

• X₃₀₁₈₁–X₃₀₂₅₇ — Frégier points

  contributed by César Eliud Lozada, December 15, 2018. Let Γ be a conic and P a fixed point on it. The sides of any right angle with vertex P cut Γ again in :g:`P’ <P’>` and :g:`P″ <P″>`. Then all the lines :g:`P’P″ <[ P’ P″ ]>` intersect in a common point Q_Γ(P). (See: Frégier’s Theorem in MathWorld). This section deals with some named conics in the plane of a triangle ABC. The point Q_Γ(P) is denoted here as the P-Frégier point of Γ. It can be proved that if Γ is a rectangular hyperbola then Q_Γ(P) lies in the line at infinity. Isogonal conjugates of these points in the infinity are also included in this section.

  Centers X₃₀₂₇₄-X₃₀₄₃₄, Endo-homothetic centers, contributed by . César Eliud Lozada, December 19, 2018. This section consists of the endo-homothetic centers of the family of triangles homothetic with the orthic triangle of a reference triangle ABC. This family is composed by the following 37 triangles:

  anti-Ascella, anti-Atik, 1st anti-circumperp, anti-Conway, 2nd anti-Conway, 3rd anti-Euler, 4th anti-Euler, anti-excenters-reflections, 2nd anti-extouch, anti-Honsberger, anti-Hutson intouch, anti-incircle-circles, anti-inverse-in-incircle, 6th anti-mixtilinear, 1st anti-Sharygin, anti-tangential-midarc, anti-Ursa minor, anti-Wasat, circumorthic, Ehrmann-side, Ehrmann-vertex, 2nd Ehrmann, 2nd Euler, 1st excosine, extangents, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, Lucas antipodal tangential, Lucas(-1) antipodal tangential, orthic, submedial, tangential, inner tri-equilateral, outer tri-equilateral, Trinh. For definitions and coordinates of these triangles, see the index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html).

• X₃₀₅₁₄–X₃₀₅₁₆ — Centers associated with intriangles and extriangles

  contributed by Clark Kimberling and Peter Moses, January 1, 2019. Following TCCT (page 196), the intriangle of a point P =\(p: q : r\) (barycentrics) is the central triangle having A-vertex

  A' = 0 : b (c q + b r cos A) : c (b r + c q) cos A

  and the extriangle of P is the central triangle having A-vertex

  A″ = -a p (c q + b r cos A)(b r + c q cos A) : a q (c q + b r cos A)(a r + c p cos B) : a p (a r + c q cos A)(a q + b p cos C)

  Properties of A'B'C' and A’’’B’’’C″:

  1. A'B'C' is inscribed in ABC, which is inscribed in A″B″C″.

  2. The locus of P for which A'B'C' is perspective to ABC is the Darboux cubic, K004; the locus of the perspector is the Thomson cubic, K002.

  3. The locus of P for which A″B″C″ is perspective to ABC is the union of the circumcircle, the lines BC, CA, AB, the line at infinity, and the cubic K004. If P is on the line at infinity, then the perspector is on the cubic K162.

  4. The locus of P for which A″B″C″ is perspective to the cevian triangle of P is the same as for property 3.

  5. The locus of P for which A'B'C' is perspective to the anticevian triangle of P is the union of the lines BC, CA, AB, and the cubic K004.

  6. A’B’C’and A″B″C″ are perspective with perspector X₆.

  7. The intriangle of X₂ is perspective to the circumsymmedial triangle at X₃.

  8. The intriangle of X₆ is perspective to the 1st Ehrmann triangle (see X₈₅₃₇ at X₁₂₀₂₇; to the Artzt triangle (see X₉₇₄₂ at X₆₇₇₆, and to the anti-Honsberger triangle at X₁₈₄.

• X₃₀₅₄₃–X₃₀₅₅₀ — Centers associated with duple triangles

  contributed by Clark Kimberling, January 1, 2019. Suppose that A'B'C' is a central triangle in the plane of a reference triangle ABC, and that barycentrics for A' are \(u : v : w\). The duple of A'B'C' is here introduced as the central triangle A″B″C″ having A″ =\(u : w : v\); for example, the duple of the excentral triangle has A″ = \(-a : c : b\). It is easy to prove that if two triangles are perspective, then their duples are perspective.

  Peter Moses (January 1, 2019) found the following perspectivities for the duple of the excentral triangle:

  ABC (TCCT 6.1): X₇₆

  medial (TCCT 6.2): X₆

  excentral (TCCT 6.7): X₃₅₀₉

  half altitude / mid-height (TCCT 6.38): X₆

  second Neuberg (MathWorld): X₅₁₁

  first Brocard (CTC): X₇₆

  anti-first-Brocard (see ETC X₅₉₃₉): X₈₇₈₂

  inner inscribed squares (MathWorld): X₆

  outer inscribed squares (MathWorld): X₆

  submedial (see ETC X₉₈₁₃): X₆

  Gemini 40: X₃₀₅₄₃

  Gemini 42: X₁₄₁

  Gemini 75: X₃₀₅₄₄

  See also the lists at X₃₀₅₄₅, X₃₀₅₄₇, and X₃₀₅₅₆.

• X₃₀₅₆₁–X₃₀₇₁₃ — Centers associated with the Gemini triangles 1 - 40

  contributed by Randy Hutson, January 9, 2019. The Gemini triangles are introduced in the preamble just before X₂₄₅₃₇.

• X₃₀₇₁₅–X₃₀₇₂₁ — Centers associated with line-reflected triangles

  contributed by Clark Kimberling, January 11, 2019). In the plane of a triangle ABC, suppose that A'B'C' is a triangle and L is a line. Let A″ be the reflection A' in L, and define B″ and C″ cyclically. The triangle A″B″C″ is here named the L-reflection of A’B’C’. If A'B'C' is a central triangle and L a central line, then A″B″C″ is a central triangle.

  Let T denote the Euler-line-reflection of ABC. Peter Moses (January 12, 2019) found that T is perspective to the following triangles, with perspectors as indicated:

  ABC: X₅₂₃

  Schroeter (anticevian triangle of X₅₂₃; see X₈₂₈₆, X₁₀₂₇₆): X₅₂₃

  tangential: X₃₀₇₁₅

  Macbeath: X₃₀₇₁₆

  orthic-of-medial (anti-6th-mixtilinear; see X₁₁₃₆₃): X₃₀₇₁₇

  5th Euler (see X₃₇₅₈: X₃₀₇₁₈

  circum-medial: X₂₃

  Gemini 44: X₂₃

  Gossard: X₃₀

  reflection of ABC in X₃: X₃₀

  infinite altitude: X₇₄

  circum-orthic: X₁₈₆

  Carnot (Johnson, the reflection of ABC in X₅): X₃₀

  Kosnita: X₁₈₆

  The triangle T is also perspective to these triangles: Euler, Trinh, 2nd Euler, 5th Euler, Artzt, anti-Artzt, tangential of tangential, anti-1st-Euler, anti-Hutson intouch, anti-incircle-circles (see X₁₁₃₆₃, orthic-of-medial, Ehrmann side-triangle.

  The locus of a point P such that the cevian triangle of P is perspective to T is the cubic pK(14618,264). The locus of P such that the anticevian triangle of P is perspective to T is the cubic pK(112,648). (Peter Moses, January 13, 2019)

  For the Nagel-line-reflection of ABC, see X₃₀₇₁₉-X₃₀₇₂₁.

• X₃₀₇₃₈–X₃₀₈₀₃ — Collineation mappings involving Gemini triangle 101

  contributed by Clark Kimberling, January 19, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, and X₂₇₃₇₈, Gemini triangles A'B'C', indexed as 101 to 111, are introduced here, given by barycentrics for A', followed by the range of associated triangle centers. Each range of centers is preceded by a preamble.

  Gemini triangle 101: A' = a^2 - b^2 - c^2 : 2 a^2 : 2 a ^2; range X₃₀₇₃₈-X₃₀₈₀₃

  Gemini triangle 102: A' = a - b - c: 2 a : 2 a; range X₃₀₈₀₈-X₃₀₈₆₉

  Gemini triangle 103: A' = - a^3 : b^3 + c^3 : b^3 + c^3; range X₃₀₈₇₈-X₃₀₉₃₉

  Gemini triangle 104: A' = - b \(c : a(b + c) : a(b + c)\); range X₃₀₉₄₂-X₃₁₀₀₇

  Gemini triangle 105: A' = - b - c : 2a + b + c : 2a + b + c; range X₃₁₀₁₄-X₃₁₀₆₃

  Gemini triangle 106: A' = - b^2 - c^2 : 2a^2 + b^2 + c^2 : 2a^2 + b^2 + c^2; range X₃₁₀₇₁-X₃₁₁₃₂

  Gemini triangle 107: A' = \(-1 : 2 : 2\); range X₃₁₁₃₃-X₃₁₁₈₁

  Gemini triangle 108: A' = 3a - b \(- c : 2(a - b - c) : 2(a - b - c)\); range X₃₁₁₈₃-X₃₁₂₃₄

  Gemini triangle 109: A' =\(1 : 2 : 2\); range X₃₁₂₃₅-X₃₁₂₈₃

  Gemini triangle 110: A' =\(2 : 1 : 1\); range X₃₁₂₈₄-X₃₁₂₈₉

  Gemini triangle 111: A' = \(-3 : 1 : 1\); range X₃₁₂₉₀-X₃₁₂₀₅

  Let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 101, as in centers X₃₀₇₃₈-X₃₀₈₀₃. Then

  m(X) = a^3 x - (a + c) (a^2 - a c + c^2) y - (a + b) (a^2 - a b + b^2) z : : ,

  and m(X) is a self-inverse mapping such that m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. Also, m(nine-point circle) = nine-point circle. The fixed points of m are X₂ and every point on the line X₃₂₅X₅₂₃, which is the isotomic conjugate of the circumcircle. Among the fixed points are X_i for these i: 2, 325, 523, 684, 850 ,3260, 3265, 3266, 3267, 20735, 30736, 30737.

• X₃₀₈₀₈–X₃₀₈₆₉ — Collineation mappings involving Gemini triangle 102

  contributed by Clark Kimberling, January 19, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 102, as in centers X₃₀₈₀₈-X₃₀₈₆₉. Then

  m(X) = (a - b - c) x + 2 b y + 2 c z : : ,

  and m(X) is a self-inverse mapping such that m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X₂ and every point on the line X₅₁₄X₆₆₁, which is the isotomic conjugate of the circumellipse {{A, B, C,X₈₈, X₁₀₀, X₁₆₂, X₁₉₀}}. Among the fixed points are X_i for these i: 2, 514, 693, 3912, 6381, 4358, 15413, 30804, 30805, 30806, 30807.

• X₃₀₈₇₈–X₃₀₉₃₇ — Collineation mappings involving Gemini triangle 103

  contributed by Clark Kimberling, January 20, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 103, as in centers X₃₀₈₇₈-X₃₀₉₃₇. Then

  m(X) = a^3 x - (a + c)(a^2 - a c + c^2) y - (a + b) (a^2 - a b + b^2) z : : ,

  and m(X) is a self-inverse mapping such that m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X₂ and every point on the line X₈₂₄X₃₀₈₇₀. Among the fixed points are X_i for these i: 2, 824, 30870, 30871, 30872, 30873, 30874, 30875, 30876, 30877.

• X₃₀₉₄₂–X₃₁₀₀₇ — Collineation mappings involving Gemini triangle 104

  contributed by Clark Kimberling, January 20, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 104, as in centers X₃₀₉₄₂-X₃₁₀₀₇. Then

  m(X) = - b c x + b (a + c) y + c (a + b) z : : ,

  and m(X) is a self-inverse mapping such that m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X₂ and every point on the line X₅₁₃X₆₉₃. Among the fixed points are X_i for these i: 2, 513, 693, 3250, 7912, 30938, 30939, 30940, 30941, 31008.

• X₃₁₀₁₄–X₃₁₀₆₃ — Collineation mappings involving Gemini triangle 105

  contributed by Clark Kimberling, January 21, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 105, as in centers X₃₁₀₁₄-X₃₁₀₆₃. Then

  m(X) = - (b + c) x + (a + 2b + c) y + (a + b + 2c) z : : ,

  and m(X) is a self-inverse mapping such that m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X₂ and every point on the line X₅₁₄X₄₀₂₄. Among the fixed points are X_i for these i: 2, 514, 4024, 4608, 6542, 31009, 31010, 31011, 31012, 31013, 31064.

• X₃₁₀₇₁–X₃₁₁₃₂ — Collineation mappings involving Gemini triangle 106

  contributed by Clark Kimberling, January 21, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 106, as in centers X₃₁₀₇₁-X₃₁₁₃₂. Then

  m(X) = - (b^2 + c^2) x + (a^2 + 2 b^2 + c^2) y + (a^2 + b^2 + 2 c^2) z : : ,

  and m(X) is a self-inverse mapping such that m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X₂ and every point on the line X₅₂₃X₂₅₂₈. Among the fixed points are X_i for these i: 2, 523, 2528, 31065, 31066, 31067, 31068, 31069, 31070.

• X₃₁₁₃₃–X₃₁₁₈₁ — Collineation mappings involving Gemini triangle 107

  contributed by Clark Kimberling, January 22, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 107, as in centers X₃₁₁₃₃-X₃₁₁₈₁. Then

  m(X) = x - 2 y - 2 z : -2 x + y - 2 z : -2 x - 2 y + z : : ,

  and m(X) is a self-inverse mapping; indeed, m(X) = reflection of X in X₂; the fixed points are X₂ and every point on the line at infinity.

• X₃₁₁₈₃–X₃₁₂₃₄ — Collineation mappings involving Gemini triangle 108

  contributed by Clark Kimberling, January 23, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 108, as in centers X₃₁₁₈₃-X₃₁₂₃₄. Then

  m(X) = (3a - b - c) x - 2(a - b + c) y - 2(a + b - c) z : :

  and m(X) is a self-inverse mapping such that m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X₂ and every point on the line X₂₄₁X₅₁₄. Among the fixed points are X_i for these i: 2, 241, 514, 650, 1323, 3008, 3676,3911,30719, 31182, 31182.

• X₃₁₂₃₅–X₃₁₂₈₃ — Collineation mappings involving Gemini triangle 109

  contributed by Clark Kimberling, January 24, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 109, as in centers X₃₁₂₃₅-X₃₁₂₈₃. Then

  m(X) = x + 2 y + 2 z : :

  and m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line.

  If P is not on the Euler line, then the (A, B, C,X₂; A',B',C',X₂) collineation image of P, where A'B'C' = Gemini triangle 109 is the intersection of lines X₂P and X₅P', where P' =reflection of P in X₃. (Randy Hutson, October 8, 2019)

• X₃₁₂₈₄–X₃₁₂₈₉ — Collineation mappings involving Gemini triangle 110

  contributed by Clark Kimberling, January 25, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 110, as in centers X₃₁₂₈₄-X₃₁₂₈₉. Then

  m(X) = 2 x + y + z : : = complement(complement(X))

  and m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. (Clark Kimberling, January 25, 2019)

  See the preamble just before X₆₆₆₆.

• X₃₁₂₉₀–X₃₁₃₀₅ — Collineation mappings involving Gemini triangle 111

  contributed by Clark Kimberling, January 25, 2018. Extending the preambles just before X₂₄₅₃₇, X₂₆₁₅₃, X₂₇₃₇₈, X₃₀₇₃₈, let m(X) denote the (A, B, C,X₂; A',B',C',X₂) collineation image of X =\(x : y : z\), where A'B'C' = Gemini triangle 111, as in centers X₃₁₂₉₀-X₃₁₃₀₅. Then

  m(X) = 3 x - y - z : : = anticomplement(anticomplement(X))

  and m(X) is collinear with X₂ and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line.

  See the preambles just before X₆₆₆₆ and X₃₁₂₈₁.

• X₃₁₃₀₆–X₃₁₃₅₂ — Perspectors involving Gemini triangles 1 to 111

  contributed by César Eliud Lozada, January 26, 2019.

  The perspector of Gemini triangles i and j is X₂ for i, j ∈ {1, 2, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111}

  That list is followed by similar lists for points other than X₂.

• X₃₁₃₅₃–X₃₁₃₇₅ — Conics associated to pairs of orthologic or parallelogic triangles

  contributed by César Eliud Lozada, January 27, 2019.

  1. Conics associated to a pair of orthologic triangles:

  Let T'=A'B'C' and T″=A″B″C″ be two orthologic triangles. Denote a’_b and a’_c the perpendicular lines from A' to A″C″ and A″B″, respectively, and build b’_c, b’_a, c’_a,c’_b cyclically. Then these six lines are tangent to a conic Φ’_t, here named the orthologic tangential-conic T’ to T”.

  Swap T' and T″, repeat the above construction and name the respective lines a”_b, a”_c, b”_c, b”_a, c”_a, c”_b. Then these six lines are tangent to another conic Φ”_t, here named the orthologic tangential-conic T” to T’.

  Let A'_b=a’_b ∩ B'C' and A'_c=a’_c ∩ B'C' and define B'_c, B'_a, C'_a, C'_b cyclically. Then these six points lie on a conic Φ’_p, here named the orthologic conic T’ to T”.

  Swap T' and T″, repeat the above construction and name the respective points A″_b, A″_c, B″_c, B″_a, C″_a, C″_b. Then these six points lie on another conic Φ”_p, here named the orthologic conic T” to T’.

  2. Conics associated to a pair of parallelogic triangles:

  Let T'=A'B'C' and T″=A″B″C″ be two parallelogic triangles. Denote a’_b and a’_c the parallel lines from A' to A″C″ and A″B″, respectively, and build b’_c, b’_a, c’_a,c’_b cyclically. Then these six lines are tangent to a conic Ψ’_t, here named the parallelogic tangential-conic T’ to T”.

  Swap T' and T″, repeat the above construction and name the respective lines a”_b, a”_c, b”_c, b”_a, c”_a, c”_b. Then these six lines are tangent to another conic Ψ”_t, here named the parallelogic tangential-conic T” to T’.

  Let A'_b=a’_b ∩ B'C' and A'_c=a’_c ∩ B'C' and define B'_c, B'_a, C'_a, C'_b cyclically. Then these six points lie on a conic Ψ’_p, here named the parallelogic conic T’ to T”.

  Swap T' and T″, repeat the above construction and name the respective points A″_b, A″_c, B″_c, B″_a, C″_a, C″_b. Then these six points lie on another conic Ψ”_p, here named the parallelogic conic T” to T’.

  If T' and T″ are orthologic (parallelogic) triangles and P is the orthologic (parallelogic) center T' to T″, then the orthologic (parallelogic) conic T' to T″ does not depend on T″. Moreover, the orthologic conic T' to T″ and the parallelogic conic T' to T″ coincide. Therefore, when T' are P are given, a more convenient name for this conic Φ’_p= Ψ’_p is P-orthoparallelogic conic of T’. A similar coincidence occurs for the tangential conics Φ’_t= Ψ’_t and therefore a better name for this conic is P-orthoparallelogic tangential-conic of T’.

  If P =\(x:y:z\) (barycentrics) then the center of the P-orthoparallelogic conic of ABC is:

  O'_p = x^2*(y+z)*((y+z)*(x^4-(y-z)^2*y*z)-x*(y^2+z^2)*(-x^2+y^2+z^2)-(y^3+z^3)*x^2) : :

  and its perspector is:

  Q'_p = x*(y+z)*F(x,y,z)*F(x,z,y) : :, where \(F(x,y,z) = (x^{2} (2 x y+x z+2 z^{2})-(y-z) \cdot (2 y ((x+z) y+z^{2})+x z (y+z)))\)

  The center of the P-orthoparallelogic tangential-conic of ABC is O'_t=:term:complement <complement>-of-P and its perspector is:

  Q'_t = (x*(2*z+y)+y*(y+z))*(x*(2*y+z)+z*(y+z)) : :

• X₃₁₃₉₃–X₃₁₅₀₂ — Centroids of curvatures

  contributed by César Eliud Lozada, February 08, 2019. Let (O₁), (O₂), (O₃) be three circles with distinct and non-collinear centers. Denote I₁, I₂, I₃ the internal centers of similitude of {(O₂), (O₃)}, {(O₃), (O₁)} and {(O₁), (O₂)} , respectively. Then the lines O₁I₁, O₂I₂, O₃I₃ concur.

  Assume that exact trilinear coordinates of the centers are O_i = ( U_i, V_i, W_i ) and their radius are \(R_{i}\) (i = 1..3). If M is the trilinear matrix of the centers then the given point of intersection Q is:

  Q = \(M^{T}\).| ρ₁ ρ₂ ρ₃ |^T, where ρ_i = 1/\(R_{i}\), i.e., ρ_i is the curvature of the circle (O_i).

  In fewer words:

  Q = Σ ρ_iU_i : Σ ρ_iV_i : Σ ρ_iW_i, i=1..3

  Then, for obvious reasons, the point Q is here named here the centroid of curvatures of the circles (O₁), (O₂), (O₃).

  Suppose a fourth circle (O₄) is added to the above configuration and let Q_ℓ be the centroid of curvatures of (O_i), (O_j), (O_k). Then the lines O_iQ_i concur at:

  Q = Σ ρ_iU_i : Σ ρ_iV_i : Σ ρ_iW_i, i=1..4

  The last extension provides a geometrical recursive construction of the centroid of curvatures of any number n ≥ 3 of circles with distinct and not collinear centers.

• X₃₁₅₂₈–X₃₁₆₀₅ — Triangles associated with Soddy circles

  contributed by César Eliud Lozada, February 19, 2019.

  Let T_aT_bT_c be the intouch triangle of ABC and (\(S_{a})\), (\(S_{b})\), (\(S_{c})\) their Soddy circles. Let A_oB_oC_o be the outer-Soddy triangle of ABC (touchpoints of Soddy circles and the circle internally tangent to them) \(and _{i}\)B_iC_i the inner-Soddy triangle of ABC (touchpoints of Soddy circles and the circle externally tangent to them). Then quadrilaterals T_bT_cC_oB_o and T_bT_cB_iC_i are cyclic. (See Antreas Hatzipolakis, Hyacinthos #28871).

  The circles (O_a) and (I_a) circumscribing B'C'C_oB_o and B'C'C_iB_i, respectively, are here named here the 2nd A-outer-Soddy circle and 2nd A-inner-Soddy circle, respectively, and triangles O_aO_bO_c and I_aI_bI_c are here named the 2nd outer-Soddy triangle and 2nd inner-Soddy triangle, respectively. Barycentric coordinates of A-vertices are:

  O_a = (-a+b+c)*a-2*S : b*(-a+b+c) : c*(-a+b+c)

  I_a = (-a+b+c)*a+2*S : b*(-a+b+c) : c*(-a+b+c)

• X₃₁₆₁₀–X₃₁₆₄₃ — Points associated with cubics pK(U’,U’): X₃₁₆₁₀-X₃₁₆₄₃

  contributed by Clark Kimberling and Peter Moses, February 20, 2019. Suppose that U =\(u : v : w\) (barycentrics) is a point in the plane of a triangle ABC. The self-inverse Gemini triangle based on U is defined as the triangle with vertices

  A' = - u : v + w : v + w, B' = - v : w + u : w + u, C' = - w : u + v : u + v .

  The triangle A'B'C' is perspective to the cevian triangle of a point X =\(x : y : z\) if and only if X lies on the cubic

  v w (y - z) (y z - x y - x z) + w u (z - x )(z x - y z - y x) + u v (x - y)(x y - z x - z y) = 0,

  this being, Bernard Gibert’s notation (Parallel Tripolars Cubics) the cubic pK(U',U'), where U' = isotomic conjugate of the crosspoint of X₂ and U; that is, U' = 1/(v+w) : 1/(w+u) : 1/(u+v). Points on this cubic, which we denote by Γ(U), include the following:

  \(1 : 1 : 1\) = X₂ = centroid

  v w : w u : u v = isotomic conjugate of U

  1/(v + w ) : 1/(w + u) : 1/(u + v) = isotomic conjugate of the complement of U

  u/(v + w ) : v/(w + u) : w/(u + v)

  v w/(v + w ) : w u/(w + u) : u v/(u + v)

  (-u + v + w)/(v + w ) : (u - v + w)/w + u) : (u + v - w)/(u + v)

  u - d : v - d : w - d, where d = \(\sqrt{v w + w u + u v}\)

  u + d : v + d : w + d, where d = \(\sqrt{v w + w u + u v}\)

  Let t1 = u^2(v+w) + v^2(w+u) + w^2(u+v) + uvw, t2 = (-u+v+w)(u-v+w)(u+v-w), and t = \(\sqrt{t1/t2}\). Then two more points on the cubic are

  u/(-u + v + w) - t : v(u - v + w) - t : w(u + v - w) - t

  u/(-u + v + w) + t : v(u - v + w) + t : w(u + v + w) - t

  If U is a triangle center, then A'B'C' is a central triangle and those ten points on Γ(U) are triangle centers. Note that the points involving square roots are nonreal for some choices of U and a,b,c.

  Let DEF be the cevian triangle of v w : w u : u v, so that D =\(0 : 1/v : 1/w\), E =\(1/u : 0 : 1/w\), F =\(1/u : 1/v : 0\). The perspector of A'B'C' and DEF is the point

  \(u(-u^{2} + v^{2} + w^{2}) : v(u^{2} - v^{2} + w^{2}) : w(u^{2} + v^{2} - w^{2})\) = U*-Ceva conjugate of U, where U* = isotomic conjugate of U.

• X₃₁₆₄₄–X₃₁₆₄₈ — Points associated with cubics pK(U’,U’): X₃₁₆₁₀-X₃₁₆₄₃

  contributed by Clark Kimberling and Peter Moses, February 20, 2019. Suppose that U =\(u : v : w\) (barycentrics) is a point in the plane of a triangle ABC. The self-inverse Gemini triangle based on U is defined as the triangle with vertices

  A' = - u : v + w : v + w, B' = - v : w + u : w + u, C' = - w : u + v : u + v .

  The triangle A'B'C' is perspective to the cevian triangle of a point X =\(x : y : z\) if and only if X lies on the cubic

  v w (y - z) (y z - x y - x z) + w u (z - x )(z x - y z - y x) + u v (x - y)(x y - z x - z y) = 0,

  this being, Bernard Gibert’s notation (Parallel Tripolars Cubics) the cubic pK(U',U'), where U' = isotomic conjugate of the crosspoint of X₂ and U; that is, U' = 1/(v+w) : 1/(w+u) : 1/(u+v). Points on this cubic, which we denote by Γ(U), include the following:

  \(1 : 1 : 1\) = X₂ = centroid

  v w : w u : u v = isotomic conjugate of U

  1/(v + w ) : 1/(w + u) : 1/(u + v) = isotomic conjugate of the complement of U

  u/(v + w ) : v/(w + u) : w/(u + v)

  v w/(v + w ) : w u/(w + u) : u v/(u + v)

  (-u + v + w)/(v + w ) : (u - v + w)/w + u) : (u + v - w)/(u + v)

  u - d : v - d : w - d, where d = \(\sqrt{v w + w u + u v}\)

  u + d : v + d : w + d, where d = \(\sqrt{v w + w u + u v}\)

  Let t1 = u^2(v+w) + v^2(w+u) + w^2(u+v) + uvw, t2 = (-u+v+w)(u-v+w)(u+v-w), and t = \(\sqrt{t1/t2}\). Then two more points on the cubic are

  u/(-u + v + w) - t : v(u - v + w) - t : w(u + v - w) - t

  u/(-u + v + w) + t : v(u - v + w) + t : w(u + v + w) - t

  If U is a triangle center, then A'B'C' is a central triangle and those ten points on Γ(U) are triangle centers. Note that the points involving square roots are nonreal for some choices of U and a,b,c.

  Let DEF be the cevian triangle of v w : w u : u v, so that D =\(0 : 1/v : 1/w\), E =\(1/u : 0 : 1/w\), F =\(1/u : 1/v : 0\). The perspector of A'B'C' and DEF is the point

  \(u(-u^{2} + v^{2} + w^{2}) : v(u^{2} - v^{2} + w^{2}) : w(u^{2} + v^{2} - w^{2})\) = U*-Ceva conjugate of U, where U* = isotomic conjugate of U.

• X₃₁₆₈₃–X₃₁₇₂₀ — Points associated with cubics pK(U*Y,U): X₃₁₆₄₄-X₃₁₆₄₈

  contributed by Clark Kimberling and Peter Moses, February 20, 2019. Suppose that U =\(u : v : w\) (barycentrics) is a point in the plane of a triangle ABC. The self-inverse Gemini triangle based on U is defined (above, in the preamble just before X₃₁₆₁₀ as the triangle with vertices

  A' = - u : v + w : v + w, B' = - v : w + u : w + u, C' = - w : u + v : u + v .

  The triangle A'B'C' is perspective to the anticevian triangle of a point X =\(x : y : z\) if and only if X lies on the cubic

  (y - z) (y z u^2 + x^2 v w) + (z - x) (z x v^2 + y^2 w u) + (x - y) (x y w^2 + z^2 u v) = 0,

  this being, in Bernard Gibert’s notation (see Special Isocubics in the Triangle Plane (Section 1.3)) the cubic pK(U*Y, U), where

  Y = 1/(v + w) : 1/(w + u) : 1/(u + v) = cevapoint of X₂ and U, and * = barycentric product.

  Points on this cubic, which we denote by Γ*(U), include the following:

  \(1 : 1 : 1\) = X₂ = centroid

  \(u : v : w\) = U

  1/(v + w ) : 1/(w + u) : 1/(u + v) = cevapoint of X₂ and U = isotomic conjugate of the complement of U

  u/(v + w ) : v/(w + u) : w/(u + v)

• X₃₁₇₂₇–X₃₁₈₄₀ — 3rd and 4th isodynamic-Dao triangles and related centers

  contributed by César Eliud Lozada, March 15, 2019. Let A_oB_oC_o be the orthic triangle of ABC. Denote by A', B', C' the 1st isodynamic points X₁₅ of AB_oC_o, BC_oA_o and CA_oB_o, respectively, and A″, B″, C″ the 2nd isodynamic points X₁₆ of AB_oC_o, BC_oA_o and CA_oB_o, respectively. Then the triangles A'B'C' and A″B″C″ are equilateral. (Dao Thanh Oai, personal communication, March 12, 2019).

  The triangles A'B'C' and A″B″C″ are here named here 3rd isodynamic-Dao and 4th isodynamic-Dao triangles, respectively. (See also triangles A*B*C* and A**B**C** constructed in Hyacinthos 28021 [Aug 11, 2018, Tran Quang Hung and Randy Hutson]). These triangles have the following properties and relations (lists follow):

• X₃₁₇₅₇–X₃₁₉₉₁ — Orthopolar circles

  contributed by César Eliud Lozada, March 22, 2019. Let :g:`T’ <T’>`=:g:`A’B’C’ << A’ B’ C’ >>` and :g:`T″ <T″>`=:g:`A″B″C″ << A″ B″ C″ >>` be two triangles. Then the orthopoles of the sidelines of :g:`T’ <T’>` with respect to :g:`T″ <T″>` and the orthopoles of the sidelines of :g:`T″ <T″>` with respect to :g:`T’ <T’>` lie on an ellipse having for center the midpoint of the orthocenters of :g:`T’ <T’>` and :g:`T″ <T″>`. When :g:`T’ <T’>` and :g:`T″ <T″>` are inscribed in concentric circles, the ellipse is a circle. Also, under certain conditions, the orthopoles may be coincident or collinear. [R. Goormaghtigh: On pairs of triangles, American Mathematical Monthly, Vol. 57, No. 3 (Mar., 1950), pp. 150-153].

  The ellipse just described is here named the orthopolar ellipse or orthopolar circle of T' and T″. This section deals with pairs of triangles inscribed in concentric circles. The appearance of (T', T″, i) in the following lists means that X_i is the center of the orthopolar circle of triangles T' and T″., contributed by . Orthopolar circles: X₃₁₇₂₇-X₃₁₈₄₀

• X₃₁₉₉₄–X₃₂₀₀₀ — Points I-Caph

  contributed by Clark Kimberling and Peter Moses, April 15, 2019. Suppose that P =\(p:q:r\) (barycentrics). The Point I-Caph of P is the point given by

  2qr + (p+q+r)p : 2rp + (p+q+r)q : 2pq + (p+q+r)r.

  See sections below for Points II-Caph, III-Caph, and IV-Caph. The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P. (The name Caph is that of a star, also known as Beta Cassiopeiae.)

Part 17

• X₃₂₀₀₁–X₃₂₀₀₇ — Points II-Caph

  contributed by Clark Kimberling and Peter Moses, April 15, 2019. Suppose that P =\(p:q:r\) (barycentrics). The Point II-Caph of P is the point given by

  2qr - (p+q+r)p : 2rp - (p+q+r)q : 2pq - (p+q+r)r.

  The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

• X₃₂₀₀₈–X₃₂₀₂₃ — Points III-Caph

  contributed by Clark Kimberling and Peter Moses, April 15, 2019. Suppose that P =\(p:q:r\) (barycentrics). The Point III-Caph of P is the point given by

  (p + q)(p + r) : (q + r)(q + p) : (r + p)(r + q)

  The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

• X₃₂₀₂₄–X₃₂₀₃₅ — Points IV-Caph

  contributed by Clark Kimberling and Peter Moses, April 15, 2019. Suppose that P =\(p:q:r\) (barycentrics). The Point IV-Caph of P is the point given by

  qr - (p + q + r)p : rp - (p + q + r)q : pq - (p + q + r)r

  The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

• X₃₂₀₃₆–X₃₂₀₄₂ — Points Chara

  contributed by Clark Kimberling and Peter Moses, April 15, 2019. Suppose that P =\(p:q:r\) (barycentrics). The Point Chara of P is the point given by

  (p - q)(p - r) : (q - r)(q - p) : (r - p)(r - q).

  Point Chara of P is the trilinear pole of the line PX₂, so that it lies on the Steiner circumellipse. (Randy Hutson, April 20, 2019)

• X₃₂₀₄₆–X₃₂₀₈₄ — Circumcenters and centroids of not-equilateral central triangles

  contributed by César Eliud Lozada, April 16, 2019.

• X₃₂₀₈₆–X₃₂₀₉₇ — Points V-Caph

  contributed by Clark Kimberling and Peter Moses, April 19, 2019. Suppose that P =\(p:q:r\) (barycentrics). The Point V-Caph of P is the point given by

  qr + 2(p + q + r)p : rp + 2(p + q + r)q : pq + 2(p + q + r)r

  The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

• X₃₂₀₉₈–X₃₂₁₀₉ — Points V-Caph

  contributed by Clark Kimberling and Peter Moses, April 19, 2019. Suppose that P =\(p:q:r\) (barycentrics). The Point VI-Caph of P is the point given by

  qr - 2(p + q + r)p : rp - 2(p + q + r)q : pq - 2(p + q + r)r

  The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

• X₃₂₁₁₀–X₃₂₁₂₇ — and X₃₂₁₁₉-X₃₂₁₂₇, Points on Walsmith rectangular hyperbola

  contributed by Peter Moses, April 20, 2019. The centers X₃₂₁₁₀-X₃₂₁₁₄ and X₃₂₁₁₉-X₃₂₁₂₇ lie on the hyperbola denoted by H₁ at K1091 (Walsmith Focal Cubic), here named the Walsmith rectangular hyperbola.

  The center of the Walsmith rectangular hyperbola is X₄₆₈, and the hyperbola passes through the vertices of the Walsmith triangles and X_i for these i: 6, 74, 110, 113, 125, 1495, 2574, 2575, 2931, 3569, 3580, 5000, 5001, 7699, 7703, 10117, 11472, 15904, 32110, 32111, 32112, 32113, 32114, 32119, 32120, 32121, 32122, 32123, 32124, 32125, 32126, 32127, 32226, 32263, 32282, 32316. The hyperbola also passes through the bicentric pair PU(4).

  Peter Moses (March 31, 2020) showed that an equation for the Walsmith rectangular hyperbola is

  f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + \(f(c,a,b,z,x,y) = 0\), where

  \(f(a,b,c,x,y,z) = b^{2} c^{2} (b^{2} - c^{2}) \cdot (-a^{2} + b^{2} + c^{2}) \cdot (2 a^{4} - a^{2} b^{2} - b^{4} - a^{2} c^{2} + 2 b^{2} c^{2} - c^{4}) x^{2} + a^{2} (b^{2} - c^{2}) \cdot (a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} - b c - c^{2}) \cdot (a^{2} - b^{2} + b c - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) y z\).

• X₃₂₁₁₅–X₃₂₁₁₈ — Incenters of not-equilateral central triangles

  contributed by César Eliud Lozada, April 19, 2019. (Includes tables of examples.)

• X₃₂₁₃₄–X₃₂₂₁₅ — of-central (not equilateral) triangles

  contributed by César Eliud Lozada, April 23, 2019. (Includes tables of examples.)

• X₃₂₂₁₇–X₃₂₂₂₉ — Points associated with the Walsmith triangle

  contributed by Peter Moses, April 27, 2019. For a definition of the Walsmith triangle, see K1091 (Walsmith Focal Cubic). See also the preamble just before X₃₂₁₁₀.

  The appearance of {i,j} in the following list means that X_j = X_i-of-Walsmith-triangle: {2,7426}, {4,32113}, {30,524}, {370,7426}, {523,1499}, {1144,7426}, {3413,542}, {3414,690}

  The circumcircle of the Walsmith triangle passes through X_i for i = 6, 23, 1316, 32221, 32222, 32224. This circle meets the Walsmith hyperbola and the Brocard circle in X₆. See X₃₂₄₄₃.

• X₃₂₂₃₇–X₃₂₃₁₇ — More centers related to Walsmith triangle

  contributed by César Eliud Lozada, May 1, 2019. This section continues the lists of centers related to Walsmith triangle given in X₃₂₂₁₇-X₃₂₂₂₉. The Walsmith triangle is directly similar to the 2nd Brocard and the 2nd orthosymmedial triangles. In both cases, the center of direct similitude is X₁₃₁₆.

  The Walsmith triangle is inversely similar to the circummedial and the 1st Ehrmann triangles with centers of inverse similitude X₃₂₃₁₄ and X₃₂₃₁₅, respectively.

• X₃₂₃₂₂–X₃₂₄₁₃ — Centers related to the Hatzipolakis-Moses triangle

  contributed by César Eliud Lozada, May 4, 2019. The Hatzipolakis-Moses triangle is defined in X₆₁₄₅.

• X₃₂₄₆₂–X₃₂₄₈₀ — Points associated with the inscribed triply bilogic triangle

  contributed by Peter Moses, May 11, 2019. The appearance of (i,j) in the following list means that X_j = ITB-isogonal conjugate of X_i, where the ITB triangle (inscribed triply bilogic triangle) is described at K003 and K1098.

• X₃₂₄₈₈–X₃₂₅₁₃ — 2nd and 3rd Vecten triangles

  contributed by César Eliud Lozada, May 14, 2019. Given a triangle ABC, build the square σ’_a = AA'_bA'_aA'_c, with the same orientation as ABC, and such that B and C lie on the lines A'_aA'_b and A'_aA'_c, respectively. Define σ’_b and σ’_c cyclically.

  Build now the square σ”_a = AA″_bA″_aA″_c, with opposite orientation of ABC, and such that B and C lie on the lines A″_aA″_b and A″_aA″_c, respectively, and construct σ”_b and σ”_c cyclically.

  A'_a and A″_a have barycentric coordinates:

  A'_a = 1/SA : 1/(SB-S) : 1/(SC-S).

  A″_a = 1/SA : 1/(SB+S) : 1/(SC+S).

  The lines AA'_a, BB`_b and CC'_a concur at X₄₈₆ and similarly the lines AA″_a, BB″_b and CC″_a concur at X₄₈₅. Moreover, the triangle bounded by the lines A'_bA'_c, B'_aB'_c and C'_aC'_b is the inner-Vecten triangle of ABC and the triangle enclosed by the lines A″_bA″_c, B″_aB″_c and C″_aC″_b is the outer-Vecten triangle of ABC. For the latter reasons, the triangle A'_aB'_bC'_c are here named here the 2nd inner-Vecten triangle of ABC and the triangle A″_aB″_bC″_c are here named the 2nd outer-Vecten triangle of ABC.

  Let A', A″ be the centers of σ’_a and σ”_a, respectively, and define (B', B″), (C', C″) cyclically. The triangles A'B'C' and A″B″C″ are here named the 3rd inner-Vecten triangle of ABC and 3rd outer-Vecten triangle of ABC, respectively. The have first vertices barycentrics coordinates:

  A' = (3*S^2+SB*SC-2*S*SW)/SA : SC-S : SB-S

  A″ = (3*S^2+SB*SC+2*S*SW)/SA : SC+S : SB+S

  Perspective triangles and perspectors:

  2nd inner-Vecten triangle: (ABC, 486), (anticomplementary, 32488), (outer-squares, 486), (inner-Vecten, 486), (3rd inner-Vecten, 486).

  2nd outer-Vecten triangle: (ABC, 485), (anticomplementary, 32489), (inner-squares, 485), (outer-Vecten, 485), (3rd outer-Vecten, 485).

  3rd inner-Vecten triangle: (ABC, 486), (medial, 32490), (outer-squares, 486), (inner-Vecten, 486), (2nd inner-Vecten, 486).

  3rd outer-Vecten triangle: (ABC, 485), (medial, 32491), (inner-squares, 485), (outer-Vecten, 485), (2nd outer-Vecten, 485).

  Orthologic triangles and orthologic centers:

  2nd inner-Vecten triangle: (3rd anti-tri-squares, 486, 32492), (Lucas(-1) reflection, 3071, 32493), (4th tri-squares, 486, 32494), (inner-Vecten, 486, 3071).

  2nd outer-Vecten triangle: (4th anti-tri-squares, 485, 32495), (Lucas reflection, 3070, 32496), (3rd tri-squares, 485, 32497), (outer-Vecten, 485, 3070).

  3rd inner-Vecten triangle: (3rd anti-tri-squares, 486, 32498), (4th tri-squares, 486, 13933), (inner-Vecten, 486, 642).

  3rd outer-Vecten triangle: (4th anti-tri-squares, 485, 32499), (3rd tri-squares, 485, 13879), (outer-Vecten, 485, 641).

  The following pairs of triangles are directly similar: {3rd inner-Vecten, Lucas(-1) central}, {3rd outer-Vecten, Lucas(+1) central}

• X₃₂₅₅₉–X₃₂₅₈₈ — Mutual polar conics

  contributed by César Eliud Lozada, May 22, 2019. Two triangles are mutually polar with respect to a conic if each edge of one is the polar of a vertex of the other. Two triangles are perspective if and only if they are mutually polar with respect to a conic.. (Lord, Eric (2013). Symmetry and Pattern in Projective Geometry, (pp 88). London: Springer).

  The conic with respect to which two perspective triangles T₁ and T₂ are mutually polar are here named the mutual polar conic of T₁ and T₂. A list of these conics involving ABC and some triangles can be seen in Wolfram’s Polar Triangle.

  If T' = A'B'C' is a central triangle perspective to ABC with perspector P =\(u:v:w\) (trilinears) then there exists a function f(a,b,c) such that AA'=f(a,b,c)*AP, BB'=f(b,c,a)*BP and CC'=f(c,a,b)*CP (in particular, if T' and ABC are homothetic then f(a,b,c) is a constant). The center O of the mutual polar conic of ABC and T' has trilinears O = u/f(a,b,c) : v/f(b,c,a) : w/f(c,a,b) (O=P if ABC and T' are homothetic). The perspector of this conic is P.

• X₃₂₆₄₀–X₃₂₇₄₁ — Centers associated with barycentric products of circumcircle-P-antipodes

  contributed by Randy Hutson, June 14, 2019. Let P =\(p : q : r\) (barycentrics). The locus of the barycentric product of circumcircle-P-antipodes is the circumconic with perspector X₆*P = \(a^{2}\)p : \(b^{2}\)q : \(c^{2}\)r.

  Let L be a line. The barycentric product of the (real or nonreal) circumcircle intercepts of L is the trilinear pole of the isogonal conjugate of the isotomic conjugate of L (or equivalently, X₆*L). These intercepts are also circumcircle-P-antipodes for all P on L.

• X₃₂₇₅₁–X₃₂₇₆₆ — Points Selected for Their Tripolar Coordinates

  contributed by Peter Moses and Clark Kimberling, June 18, 2019. Homogeneous tripolar coordinates (henceforth simply tripolars) for a point X are any triple \(x : y : z\) of numbers (or functions of a,b,c) that are respectively proportional to the distances |AX|, |BX|, |CX|. A point X and its circumcircle-inverse have identical tripolars. The appearance of i, j; f(a,b,c) in the following table means that X_i and X_j are such a pair, with tripolars \(f(a,b,c,) : f(b,c,a) : f(c,a,b)\). (Table omitted.)

  If X has tripolars \(x: y : z\), then barycentrics for X are

  a^2 (D SA + T S) : : and a^2( D SA - T S) : : , where SA = b c cos A, S = 2 area(ABC), T = b^2 y^2 + c^2 z^2 - a^2 x^2, and

  D = \(\sqrt{(a x + b y + c z)(- a x + b y + c z)(a x - b y + c z)(a x + b y - c z)}\).

  For details, see X₅₀₀₂. Further developments are given in Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines and Albrecht Hess, Transforming Tripolar into Barycentric Coordinates

• X₃₂₇₇₁–X₃₂₇₈₄ — Centers of degree 3 with coefficients in {-1,0,1}

  contributed by Peter Moses, June 20, 2019. Each center in this section has barycentrics of the form p(a,b,c) : : , where p(a,b,c) is a polynomial homogeneous of degree 3 in a,b,c, with coefficents in the set {-1, 0 , 1}. For more of these, see also X₃₂₈₄₂-X₃₂₈₆₆, X₃₂₉₁₁-X₃₂₉₅₀, and X₃₃₀₆₄-X₃₃₁₇₅.

• X₃₂₇₈₅–X₃₂₈₄₁ — Points with first barycentric of type h f(a,b,c) + k S: X₃₂₇₈₅-X₃₂₈₄₁

  contributed by Clark Kimberling, June 20, 2019.

  Centers X₃₂₇₈₅ - X₃₂₇₉₀ … … . of form h a^2 + k S : :

  Centers X₃₂₇₉₁ - X₃₂₈₀₄ … … . of form h b c + k S : :

  Centers X₃₂₈₀₄ - X₃₂₈₄₁ … … . of form h SA^2 + k S : :

• X₃₂₈₄₂–X₃₂₈₆₆ — Centers of degree 3 with coefficients in {-1,0,1}

  :contributed by Peter Moses, June 21, 2019: Each center in this section has barycentrics of the form p(a,b,c) : : , where p(a,b,c) is a polynomial homogeneous of degree 3 in a,b,c, with coefficents in the set {-1, 0 , 1}. For more of these, see also X₃₂₇₇₁-X₃₂₇₈₄, X₃₂₉₁₁-X₃₂₉₅₀, and X₃₃₀₆₄-X₃₃₁₇₅.

• X₃₂₈₆₇–X₃₂₈₉₃ — Points h SB SC + k S^2 on the line X₂X_39

  contributed by Clark Kimberling, June 20, 2019.

• X₃₂₉₁₁–X₃₂₉₅₀ — Centers of degree 3 with coefficients in {-1,0,1}

  contributed by Peter Moses, June 21, 2019. Each center in this section has barycentrics of the form p(a,b,c) : : , where p(a,b,c) is a polynomial homogeneous of degree 3 in a,b,c, with coefficents in the set {-1, 0 , 1}. For more of these, see also X₃₂₇₇₁-X₃₂₇₈₄, X₃₂₈₄₂-X₃₂₈₆₆, and X₃₃₀₆₄-X₃₃₁₇₅.

• X₃₂₉₅₁–X₃₃₀₆₃ — Points h SB SC + f(a,b,c) on the Euler line

  contributed by Clark Kimberling, June 21, 2019:

• X₃₃₀₆₄–X₃₃₁₇₅ — Centers of degree 3 with coefficients in {-1,0,1}

  contributed by Peter Moses, June 20, 2019: Each center in this section has barycentrics of the form p(a,b,c) : : , where p(a,b,c) is a polynomial homogeneous of degree 3 in a,b,c, with coefficents in the set {-1, 0 , 1}. For more of these, see also X₃₂₇₇₁-X₃₂₇₈₄, X₃₂₈₄₂-X₃₂₈₆₆, and X₃₂₉₁₁-X₃₂₉₅₀.

• X₃₃₁₈₀–X₃₃₂₈₀ — Points on the Euler line

  contributed by Clark Kimberling, June 22, 2019. This section consists of points having barycentrics of the form h a^2 SA + k SB SC + m (a^4 + b^4 + c^4) : : .

• X₃₃₃₃₈–X₃₃₅₀₃ — Half-squares and half-diamonds triangles

  contributed by César Eliud Lozada, June 27, 2019. In a triangle ABC, build the triangles T_a = AB'C', T_b = BC'A' and T_c = CA'B', all isosceles with the same vertical angle θ in A, B, C, respectively. Barycentrics coordinates of A' are

  A' = (a^2*sin(θ) + S)/a : b*sin(C-θ) : c*sin(B-θ)

  and cyclically for B' and C'.

  There are two triads of such triangles, according as T_a, T_b, T_c have all the same or opposite orientation as ABC (this fact can be translated by using ±θ). Let A'B' C' and A″B″C″ be the respective resulting triangles. Then

  A'B'C' and A″B″C″ are both perspective to ABC with perspectors Q = a*csc(A - θ) : : (barycentrics). These perspectors lie on the Kiepert hyperbola of ABC and their isogonal conjugates satisfy

  \(OQ^{-1}\) = OK*cos(θ)*cos(ω)/cos(θ + ω), where O=X₃, K=X₆ and ω is the Brocard angle of ABC. A'B'C' and A″B″C″ are perspective with perspector Q* = cos(A)*cos(θ)+cos(B)*cos(C) : :. Q* lies on the Euler line of ABC and OQ* = OH/(1+2*cos(θ)), where H=X₄. ABC is orthologic to both A'B'C' and A″B″C″. The orthologic centers A'B'C'→ABC and A″B″C″→ABC are both Q and the reciprocal orthologic centers lie also on the Kiepert hyperbola of ABC.
  Two particular cases are considered here: θ = ±π/2 and θ = ±π/3. For the first case, T_a is a half-square (and similarly for T_b, T_c) and, for the second case, these three triangles are equilateral and we can regard T_a as a half-diamond. These names are introduced here for the triangles A'B'C' and A″B″C″. Their A-vertex barycentrics coordinates are

  1st half-squares triangle, with A' = -(a^2-S)/a : b*cos(C) : c*cos(B) = -(a^2+S) : SC : SB

  2nd half-squares triangle, with A″ = -(a^2+S)/a : b*cos(C) : c*cos(B) = -(a^2-S) : SC : SB

  1st half-diamonds triangle, with A' = -(\(\sqrt{3}\)*a^2-2*S)/(2*a) : b*cos(C-π/6) : c*cos(B-π/6) = -(\(\sqrt{3}\)*a^2-2*S) : \(\sqrt{3}\)*SC+S : \(\sqrt{3}\)*SB+S

  2nd half-diamonds triangle, with A″ = -(\(\sqrt{3}\)*a^2+2*S)/(2*a) : b*cos(C+π/6) : c*cos(B+π/6) = -(\(\sqrt{3}\)*a^2+2*S) : \(\sqrt{3}\)*SC-S : \(\sqrt{3}\)*SB-S
  Moreover, when θ = π/3, the centers A_o, B_o, C_o of the equilateral triangles T_a, T_b and T_c are vertices of an equilateral triangle. These triangles are here named the 1st and 2nd diamonds-central triangles. They have center X₂ and A-vertices

  1st half-diamonds-central triangle: A'_o = -(a^2+2*\(\sqrt{3}\)*S) : SC-\(\sqrt{3}\)*S : SB-\(\sqrt{3}\)*S

  2nd half-diamonds-central triangle: A″_o = -(a^2-2*\(\sqrt{3}\)*S) : SC+\(\sqrt{3}\)*S : SB+\(\sqrt{3}\)*S

  Also, for any value of θ, the 1st or 2nd Fermat points of T_a, T_b, T_c are the vertices of other equilateral triangles.

  A complete list of triangles and centers related with the six triangles above defined can be seen here (half_squares_lists.pdf).

• X₃₃₅₁₃–X₃₃₅₁₆ — Centers of bianticevian conics

  contributed by Vu Thanh Tung and Vu Quoc My, July 4, 2019. Suppose that P and Q are points not on a sideline of a triangle ABC. The conic section that passes through P, Q, and the vertices of the anticevian triangle of P also passes through the vertices of the anticevian triangle of Q. This conic was introduced by Randy Hutson. Barry Wolk found coordinates for the center, and Paul Yiu and Francisco Javier García Capitán found other properties. See Hyacinthos (#21109 and related posts), July 2012.

  If P =\(p : q : r\) and U =\(u : v : w\) (barycentrics), then the two anticevian triangles have vertices

  P_a = \(-p : q : r\), P_b =\(p :-q : r\), P_c = p : q -r and U_a = \(-u : v : w\), U_b=\(u : -v : w\), U_c =\(u : v : -w\),

  and the conic, which passes through P, Q, P_a, P_b, P_c, U_a, U_b, U_c, has center 1/(q^2 w^2 - r^2 v^2) : 1/(r^2 u^2 - p^2 w^2) :1/(p^2 v^2 - q^2 w^2).

• X₃₃₅₃₇–X₃₃₅₄₃ — Points on the Stammler reflection hyperbola

  :contributed by Peter Moses, July 11, 2019. The name Stammler reflection hyperbola, denoted by SRH, refers to the reflection of the Stammler hyperbola in X₃. Let T be the hexyl triangle. Then SRH is the circumhyperbola of T that passes through the orthocenter and nine-point center of T; i.e., SRH = ABCHN-of-T. Moreover,

  SRH = Feuerbach hyperbola of tangential triangle of Thomson triangle

  SRH = isogonal conjugate of the Euler line of ABC wrt the tangential triangle of Thomson triangle

  SHR has center X₇₄, perspector X₂₆₉₃, and equation

  b^2 c^2 (b^2-c^2) ((3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) x^2 + 4 a^2 (-a^2+b^2+c^2) y z) + (cyclic) = 0

  SRH passes through the vertices of these triangles:

  hexyl (TCCT 6.36)

  excentral triangle of Thomson triangle (CTC Q113)

  tangential triangle of Thomson triangle

  anti-Hutson intouch triangle (see X₁₁₃₆₃)

  SRH passes through X_i for these i: 3, 40, 64, 1350, 2574, 2575, 5373, 5646, 9914, 10117, 10620, 11472, 12163, 12301, 12302, 12307, 14926, 15622, 16010, 19376, 22549, 32345, 33535, 33536, 33537, 33538, 33539, 33540, 33541, 33542, 33543

  Suppose that P =\(p : q : r\) (barycentrics). Define the Gibert-Euler→SRH mapping by GE(P) = a^2(8 b^2 c^2 p^2 - 2 c^2 (a^2 - c^2) q^2 - 2 b^2 (a^2 - b^2) r^2 - (a^2 (a^2 - b^2 - c^2) - 4 b^2 c^2) q r - b^2 (a^2 - b^2 - 7 c^2) r p - c^2 (a^2 - 7 b^2 - c^2) p q) : : . If P is on the Euler line, then GE(P) is on SRH.

• X₃₃₆₀₂–X₃₃₆₂₇ — Fermat-Dao-Nhi triangles and related centers

  contributed by César Eliud Lozada, July 19, 2019. Dao Thanh Oai and Hoang Le Ngoc Nhi have presented the following configurations:

  Let ABC be a triangle with centroid G and outer-Fermat (or inner-Fermat) triangle A_fB_fC_f.

  Let A* = reflection of G in A_f; let A₁ = reflection of A* in A. Define B₁ and C₁ cyclically. Let A* = reflection of G in A; let A₂ = reflection of A* in A_f . Define B₂ and C₂ cyclically. Let A* = reflection of A in G; let A₃ = midpoint of A*A_f . Define B₃ and C₃ cyclically. Let A* = reflection of A_f in G; let A₄ = midpoint of A*A. Define B₄ and C₄ cyclically. Then triangles A₁B₁C₁, A₂B₂C₂, A₃B₃C₃, A₄B₄C₄ are equilateral and perspective to ABC.

  Added by César Lozada:

  For the outer-Fermat triangle, barycentrics coordinates of A₁…A₄ are:

  A₁ = -7*S-3*\(\sqrt{3}\)*a^2 : 2*S+3*\(\sqrt{3}\)*SC : 2*S+3*\(\sqrt{3}\)*SB

  A₂ = -5*S-3*\(\sqrt{3}\)*a^2 : 4*S+3*\(\sqrt{3}\)*SC : 4*S+3*\(\sqrt{3}\)*SB

  A₃ = 4*S-3*\(\sqrt{3}\)*a^2 : -5*S+3*\(\sqrt{3}\)*SC : -5*S+3*\(\sqrt{3}\)*SB

  A₄ = 8*S-3*\(\sqrt{3}\)*a^2 : -S+3*\(\sqrt{3}\)*SC : -S+3*\(\sqrt{3}\)*SB

  For the inner-Fermat triangle, just replace S → -S in these coordinates.

  The preceding triangles are here named here the 1st, 2nd, 3rd and 4th outer-Fermat-Dao-Nhi triangles, respectively, and their correspondents built from the inner-Fermat triangle are here named the 1st, 2nd, 3rd and 4th inner-Fermat-Dao-Nhi triangles, respectively. All these triangles have center X₂.

  Observe that given A-barycentrics have all the form A_{λ, μ} = λ*S - a^2 : μ*S + SC : μ*S + SB, where λ, μ are real numbers. It results that, for all real numbers (λ, μ), the triangle 𝔽(λ, μ) = A_{λ, μ}B_{λ, μ}C_{λ, μ} is perspective to each triangle in the following set: {ABC, anti-Artzt, 1st anti-Brocard, anti-McCay, anticomplementary, Artzt, 1st Brocard-reflected, 1st Brocard, inner-Fermat, outer-Fermat, 1st half-diamonds-central, 2nd half-diamonds-central, 1st half-diamonds, 2nd half-diamonds, 1st half-squares, 2nd half-squares, McCay, medial, inner-Napoleon, outer-Napoleon, 1st Neuberg, 2nd Neuberg, inner-Vecten, outer-Vecten}. Also 𝔽(λ, μ) and 𝔽(λ’, μ’) are perspective for any combination of (λ, μ, λ’, μ’). The perspector of ABC and 𝔽(λ, μ) has barycentric coordinates Q = (μ*S+SB)*(μ*S+SC) : :, not dependending on λ. Q lies on the Kiepert hyperbola of ABC.

  With the last notation we have: A₁B₁C₁ = 𝔽(-7*k, 2*k), A₂B₂C₂ = 𝔽(-5*k, 4*k), A₃B₃C₃ = 𝔽(4*k, -5*k), A₄B₄C₄ = 𝔽(8*k, -k), where k = ±\(\sqrt{3}\)/9 (plus and minus signs for outer- and inner- triangles, respectively). Similarly: 1st/2nd half-diamonds triangles = 𝔽( ±2*\(\sqrt{3}\)/3, ±\(\sqrt{3}\)/3 ), 1st/2nd half-diamonds-central triangles = 𝔽( ∓2*\(\sqrt{3}\), ∓\(\sqrt{3}\) ), 1st/2nd half-squares triangles = 𝔽( ∓1, 0 ), outer-/inner-Fermat = 𝔽( 0, ±\(\sqrt{3}\)/3 ).

  The condition for 𝔽(λ, μ) being equilateral is |λ - μ| = \(\sqrt{3}\).

  Note that every original construction is made from two reflections. Going further, if homothecies are used instead of reflections then configurations 1 to 4 can be combined and generalized by writing:

  <blockquote

  Let ABC be a triangle with centroid G and outer-Fermat (or inner-Fermat) triangle A_fB_fC_f.

  Let A* be the homothecy of G with factor p and center A, 𝔸₁ the homothecy of A* with factor q and center A_f. Build 𝔹₁ and ℂ₁ cyclically. Then, for each value of p∉{0, 3} there exist two values of q∈{1/p, 2/(3-p)} such that 𝔸₁𝔹₁ℂ₁ is equilateral and perspective to ABC. Let A* be the homothecy of G with factor p and center A_f, 𝔸₂ the homothecy of A* with factor q and center A. Build 𝔹₂ and ℂ₂ cyclically. Then, for each value of p∉{0, 3/2} there exist two values of q∈{1/p, 1/(3-2*p)} such that 𝔸₂𝔹₂ℂ₂ is equilateral and perspective to ABC.

• X₃₃₆₂₈–X₃₃₆₃₆ — Perspectors of triangles inscribed in the circumcircle

  contributed by Clark Kimberling and Peter Moses, July 21, 2019. Suppose that P =\(p : q : r\) is a point. The points q - r : r - p : p - q and 2p - q - r : 2q - r - p : 2r - p - q clearly lie on the line at infinity, so that their isogonal conjugates, \(g(P) = a^{2}/(q-r)\): b^2/(r-p) : c^2/(p-q) and \(h(P) = a^{2}/(2p-q-r)\): b^2/(2q-r-p) : c^2/(2r-p-q), lie on the circumcircle. Thus, if T is a central triangle Then g(T) and h(T) are central triangles inscribed in the circumcircle. Suppose that X =\(x : y : z\) is a triangle center. Let T(1) be the central triangle with A-vertex \(-x : y : z\), and let T(2) be the central triangle with A-vertex \(0 : 1/y : 1/z\).

  Theorem: the triangles g(T(1)) and h(T(2)) are perspective, and their perspector is

  M(X) = a^2(-3x+y+z)/(y+z) : b^2(-3y+z+x)/(z+x) : c^2(-3z-x+y)/(x+y).

  If X is on the circumcircle, then M(X) is on the line X₆X', where X' is the X₃-cross conjugate of X. If X is on the line at infinity, then M(X) = X₆.

  If X is on the Kiepert circumhyperbola, KCH, of the anticomplementary triangle, then M(X) is on the Brocard axis. Points on KCH include X_i for these i: {1, 2 , 20, 63, 147, 194, 368, 487, 488, 616, 617, 627, 628, 1670, 1671, 1764, 2128, 2582, 2583, 2896, 3413, 3414, 6194, 6462, 6463, 7616, 8591, 8782, 9742, 10336, 11148, 13174, 13678, 13798, 16552, 16563, 17147, 18301, 18596, 20371, 21378, 30562, 30564, 30579, 33404, 33405, 33608, 33609, 33610, 33611, 33612, 33613}

  See César Lozada, Perspectivities involving the g and h mappings (g(T)_h(T).pdf).

• X₃₃₇₆₀–X₃₃₇₈₀ — Perspectors T1(X_i, X_j) involving triangles inscribed in a circumconic

  contributed by Clark Kimberling and Peter Moses, August 4, 2019. Suppose that P =\(p : q : r\) and U =\(u : v : w\) (barycentrics) are points. Define M(P,U) = p/(v - w) : q/(w - u) : r/(u - v), so that M(P,U) lies on the circumconic p/x + q/y + r/z = 0.

  Let T₁ be the anticevian triangle of P, given by vertices

  A' = \(-p : q : r\)

  B' =\(p : - q : r\)

  C' =\(p : q : -r\)
  Let T₂ be the anticevian triangle of U, given by vertices

  A″ = \(-u : v : w\)

  B″ =\(u : - v : w\)

  C″ =\(u : v : -w\)

  Let M(P,T₂) denote the triangle with vertices M(P,A″), M(P,B″), M(P,C″). The triangles T₁ and M(T₂) are perspective, and their perspector is given by

  T₁(P,U) = p*(u^2 + uv + uw - vw) : q*(v^2 + vw + vu - wu) : r*(w^2 + wu + wv - uv)

  The appearance of (i,j,k) in the following list means that T₁(X_i,X_j) = X_k. (List omitted here.)

• X₃₃₇₈₁–X₃₃₇₈₂ — Perspectors T2(X_i, X_j) involving triangles inscribed in a circumconic

  contributed by Clark Kimberling and Peter Moses, August 4, 2019. Suppose that P =\(p : q : r\) and U =\(u : v : w\) are points Define M(U) = p/(v - w) : q/(w - u) : r/(u - v), so that M(U) lies on the circumconic p/x + q/y + r/z = 0

  Let T₁ be the triangle with vertices

  A' = \(-p : q : r\)

  B' =\(p : - q : r\)

  C' =\(p : q : -r\)
  Let T₂ be the triangle with vertices

  A″ =\(0 : y : z\)

  B″ =\(x : 0 : z\)

  C″ =\(x : y : 0\)

  The triangles T₁ and M(T₂) are perspective, and their perspector is given by

  T₂(P,X) = p*(x^2 y^2 + x^2 z^2 - y^2 z^2) : q*(y^2 z^2 + y^2 x^2 - z^2 x^2) : r*(z^2 x^2 + z^2 y^2 - x^2 y2)

  The appearance of (i,j,k) in the following list means that T₂(X_i,X_j) = X_k. (List omitted here.)

• X₃₃₇₉₀–X₃₃₈₀₉ — Perspectors T3(X_i, X_j) involving triangles inscribed in a circumconic

  contributed by Clark Kimberling and Peter Moses, August 4, 2019. Suppose that P =\(p : q : r\) and U =\(u : v : w\) are points Define M(U) = p/(v - w) : q/(w - u) : r/(u - v), so that M(U) lies on the circumconic p/x + q/y + r/z = 0.

  Let T₁ be the triangle with vertices

  A' = \(-p : q : r\)

  B' =\(p : - q : r\)

  C' =\(p : q : -r\)
  Let T₂ be the triangle with vertices

  A″ =\(0 : z : -y\)

  B″ = \(-z : 0 : x\)

  C″ =\(y : -x : 0\)

  The triangles T₁ and M(T₂) are perspective, and their perspector is given by

  T₃(P,X) = p*(y^2 + z^2 - x^2) : q*(z^2 + x^2 - y^2) : r*(x^2 + y^2 - z^2)

  The appearance of (i,j,k) in the following list means that T₃(X_i,X_j) = X_k. (List omitted here.)

• X₃₃₈₁₆–X₃₃₈₄₁ — 4th degree polynomial centers on the Euler line

  contributed by Peter Moses, August 5, 2019

• X₃₃₉₀₄–X₃₃₉₂₂ — Points on the line at infinity

  contributed by Clark Kimberling, August 13, 2019. Suppose that U =\(u : v : w\) =\(u(a,b,c) : v(a,b,c) : w(a,b,c)\) is a point on the line L at infinity, which consists of all points X =\(x : y : z\) : =\(x(a,b,c) : y(a,b,c): z(a,b,c)\) that satisfy \(x+y+z = 0\). Suppose also that D =\(d : e : f\) is a point, and define

  g(D) =\(u(d,e,f) : v(d,e,f) : w(d,e,f)\).

  Clearly, g(D) lies on L. For example, if D = U = b-c : c-a : a-b, then

  \(g(D) = c-a - (a-b)\): a-b - (b-c) : b-c - (c-a) = 2a-b-c : 2b-c-a : 2c-a-b, and g(g(D)) = D; also,

  if D = 2a-b-c : 2b-c-a : 2c-a-b, then \(g(D) = D\).

  In those examples, a,b,c can be replaced by \(a^{m}\), \(b^{m}\),\(c^{m}\) for any nonzero real number, so that that following family of triangle centers, all on L, can be regarded as a basis for generating, by composition, many more centers on L.

Part 18

• X₃₄₀₁₆–X₃₄₀₈₈ — Perspectors involving Talitha triangles

  contributed by Clark Kimberling and Peter Moses, August 19, 2019. Suppose that P =\(p : q : r\) is a point. The P-Talitha triangle of X, denoted by TT(P,X), is the triangle A'B'C' is defined by its vertices:

  A' = -p/(y - z) : q/(z + x) : -r/(x + y)

  B' = -p/(y + z) : -q/(z - x) : r/(x + y)

  C' = p/(y + z) : -q/(z + x) : -r/(x - y).

  The triangle A'B'C' is inscribed in the circumconic -p y z + q z x + r x y = 0.

  The name: Talitha is a star on the front paw of Ursa Major.

• X₃₄₁₇₈–X₃₄₁₉₃ — P-vertex conjugate of P, for P on the line at infinity

  contributed by Randy Hutson, August 31, 2019. Let P be a point on the line at infinity. The locus of the P-vertex conjugate of P, as P varies, is a quartic which is the isogonal conjugate of the anticomplementary circle. This quartic passes through the vertices of ABC and the tangential triangle, the circular points at infinity, and centers X₃₄₄₆, X₃₄₄₇, X₉₂₁₇, X₂₂₂₅₉, X₃₄₁₃₀, X₃₄₁₇₈, X₃₄₁₇₉, X₃₄₁₈₀, X₃₄₁₈₁, X₃₄₁₈₂, X₃₄₁₈₃, X₃₄₁₈₄, X₃₄₁₈₅, X₃₄₁₈₇, X₃₄₁₈₉, X₃₄₁₉₀, X₃₄₁₉₁ and X₃₄₁₉₂. If ABC is acute, the quartic is a closed curve. If ABC is obtuse, it meets the line at infinity at the isogonal conjugates of the anticomplements of PU(4) (the circumcircle intercepts of the anticomplementary circle).

  Peter Moses gives the equation for the quartic as:

  c^6 x^2 y^2 + b^2 c^2 (a^2 + b^2 + c^2) x^2 y z + a^2 c^2 (a^2 + b^2 + c^2) x y^2 z + b^6 x^2 z^2 + a^2 b^2 (a^2 + b^2 + c^2) x y z^2 + a^6 y^2 z^2 = 0 See Hyacinthos 29443.

• X₃₄₃₄₁–X₃₄₃₄₄ — Frégier points

  contributed by Clark Kimberling and Peter Moses, September 30, 2019. Suppose that P =\(p : q : r\) is a point in the plane of a triangle ABC. Suppose further that p,q,r are distinct homogeneous functions of a,b,c. The permutation ellipse of P, denoted by E(P), is the ellipse that passes through the six points \(p : q : r\),\(q : r : p\),\(r : p : q\),\(p : r : q\),\(q : p : r\),\(r : q : p\), given by

  (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  The centroid, X₂, is the center and the perspector of E(P).

  E(X₉₉) = Steiner circumellipse

  E(X₁₁₅) = Steiner inellipse

  E(X₁₄₈) = Steiner circumellipse of anticomplementary triangle.
  E(X₁) passes through X_i for i = 1, 3679, 24338, 24345, 24411.

  E(X₆) passes through X_i for i = 6, 599, 24281, 24289, 30229.

  E(X₁₀) passes through X_i for i = 10, 551, 24348, 25382.

  E(X₉₈) passes through X_i for i = 98, 6039, 6040, 6054.

  E(X₆₉) passes through X_i for i = 69, 1992, 30225, 30226, 30227, 30228, 34341, 34342, 34343, 34344.

  E(X₁₂₅) passes through X_i for i = 125, 5642, 16278.

  E(X₆₂₀) passes through X_i for i = 620, 4422, 5461, 17044, 23583, 27076

  Further discussion of permutation ellipses is given in the preamble just before X₃₅₀₂₅.

  The Frégier point of P is the point P' shown at Frégier’s Theorem. Suppose that P =\(p : q : r\) lies on the Steiner circumellipse. Then the Frégier point of P is given by

  \(F(P) = (-5 a^{2} + b^{2} + c^{2}) q^{2} r^{2} + p^{2} (r^{2} (a^{2} + b^{2} - 5 c^{2}) + q^{2} (a^{2} - 5 b^{2} + c^{2})) + p q r ((-7 a^{2} + 5 b^{2} + 5 c^{2}) p + (5 a^{2} + 5 b^{2} - 7 c^{2}) q + (5 a^{2} - 7 b^{2} + 5 c^{2}) r)\): : .

  The locus of F(P) for P on the Steiner circumellipse is the permutation ellipse E(X₆₉). Selected points on the Steiner circumellipse and the associated Frégier points on E(X₆₉) are shown here:

  X₉₉ → X₆₉, X₆₇₁ → X₁₉₉₂, X₁₉₀ → X₃₀₂₂₅, X₂₉₀ → X₃₀₂₂₆,

  X₆₄₈ → X₃₀₂₂₇, X₆₆₄ → X₃₀₂₂₈, X₆₇₀ → X₃₂₃₄₁, X₉₀₃ → X₃₄₃₄₂,

  X₃₂₂₇ → X₃₄₃₄₃, X₃₂₂₈ → X₃₄₃₄₄; some of these are noted at X₃₀₂₂₅.

  An equation for the ellipse E(X₆₉), that is, Frégier image of the Steiner circumellipse, is

  (a-b-c) (a+b-c) (a-b+c) (a+b+c) x^2+(3 a^4-2 a^2 b^2+3 b^4-2 a^2 c^2-2 b^2 c^2+3 c^4) y z + (cyclic) = 0.

  An equation for the ellipse E(X₆), which is the Frégier image of the Steiner inellipse, is

  (a^2 b^2+a^2 c^2+b^2 c^2) x^2 - (a^4+b^4+c^4) y z + (cyclic) = 0.

  In general, the Frégier image of an ellipse is an ellipse; specifically, it is the dilation of the first ellipse by the following factor:

  (semiMajor^2-semiMinor^2)/(semiMajor^2+semiMinor^2).

• X₃₄₄₂₆–X₃₄₄₄₉ — Vertex conjugates

  contributed by Clark Kimberling and Peter Moses, October 11, 2019. If P =\(p : q : r\) (barycentrics), then (P-vertex conjugate of P) = isogonal conjugate of anticomplement of isogonal conjugate of P, denoted and given by

  V(P) = a^2 / ( -a^2 q r + b^2 r p + c^2 p q) : : .

  If P is on the circumcircle, then V(P) = P.

  If P is on the Steiner circumellipse, then V(P) lies on the circumconic given by

  a^2 c^4 x y+b^2 c^4 x y+a^2 b^4 x z+b^4 c^2 x z+a^4 b^2 y z+a^4 c^2 y z = 0 ,

  which passes through X_i for these i: 99, 1576, 1634, 3455, 9468, 34067. This is the circumconic with perspector X₃₀₅₁ and center X₃₄₄₅₂.

  If P is on the Kiepert hyperbola, then V(P) lies on the circumconic given by

  a^4 c^4 x y-b^4 c^4 x y-a^2 c^6 x y+b^2 c^6 x y-a^4 b^4 x z+a^2 b^6 x z-b^6 c^2 x z+b^4 c^4 x z-a^6 b^2 y z+a^4 b^4 y z+a^6 c^2 y z-a^4 c^4 y z = 0 ,

  which passes through X_i for these i: 3, 25 , 32, 98, 184, 228, 878, 1402, 1410, 1799, 2200, 2351, 2353, 3425, 3437, 3438, 3439, 3442, 3443, 3455, 3456, 3504, 6401, 6402, 8825, 8858, 8884, 10547, 14600, 14908, 17970, 22381, 22455, 23716, 23717, 33581. This is the circumconic with perspector X₃₀₄₉ and center X₁₇₄₂₃.

  If P is on the Jerabek hyperbola, then V(P) also lies of the Jerabek hyperbola, as in X_i for i = 34435-34440.

  If P is on the Feuerbach hyperbola, then V(P) is on the circumconic given by

  a^3 c^3 x y-a^2 b c^3 x y+a b^2 c^3 x y-b^3 c^3 x y-a c^5 x y+b c^5 x y-a^3 b^3 x z+a b^5 x z+a^2 b^3 c x z-b^5 c x z-a b^3 c^2 x z+b^3 c^3 x z-a^5 b y z+a^3 b^3 y z+a^5 c y z-a^3 b^2 c y z+a^3 b c^2 y z-a^3 c^3 y z = 0,

  which passes through X_i for these i: 3, 28, 48, 56, 104, 603, 911, 963, 1333, 1436, 1437, 1444, 1472, 1791, 1811, 2196, 2217, 3417, 3418, 3420, 3433, 3435, 7053, 10623, 15617, 17971, 20779, 23086, 32658, 34121, 34125, 34250. This is the circumconic with perspector X₂₂₃₈₃ and center X₃₄₄₆₇.

  In general, if P is on the circumconic with perspector \(u : v : w\), then V(P) is on the circumconic with perspector a^4 (c^2 v+b^2 w) : : . If CC is a circumconic on the Lemoine line (which passes through X_i for i = 187, 237, 351, 352, 512, 647, 649, …}, then V maps CC to CC.

• X₃₄₄₅₃–X₃₄₄₆₆ — Points on the Apollonius circle

  based on notes from Peter Moses, October, 2019. The excircles are tangent to the Apollonius circle and the nine-point circle. The centers of the excircles, therefore, lie on an ellipse, here named the Moses-Apollonius ellipse, that is the locus of the center of a (dynamic) circle tangent to the Apollonius and nine-point circles. An equation for this ellipse follows:

  b^2*c^2*(a + b - c)*(a - b + c)*(b + c)^2*x^2 - 2*a^2*b*c*(a + b + c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*y*z + (cyclic) = 0

  The ellipse passes through the vertices of the vertices of the excentral triangle and X_i for these i: 5400, 34460, 34461, 34462, 34463, 34464, 34465 and has

  center X₃₄₄₆₆

  perspector X₂₀₅₁

  foci X₅ and X₉₇₀

  eccentricity \(\sqrt{1-S/(R s)}\) = OI/R = |X₁X₃|/R

• X₃₄₄₈₈–X₃₄₅₀₀ — Bevan-antipodal triangle and related centers

  contributed by César Eliud Lozada, October 15, 2019. Let V be the Bevan point X₄₀ of ABC and V_a, V_b, V_c the antipodes of V on the circumcircles of VBC, VCA and VAB, respectively. The triangle V_aV_bV_c will be named here the Bevan-antipodal triangle of ABC. (see Angel Montesdeoca, HG141019 and Antreas Hatzipolakis Hyacinthos #29638.

  Barycentric A-vertex coordinates of the Bevan antipodal triangle are: -a/(-a+b+c) : b/(a-b+c) : c/(a+b-c)

  The Bevan-antipodal triangle is perspective to the following triangles with the given perspectors:

  (ABC, 57), (ABC-X3 reflections, 3182), (anti-tangential-midarc, 1), (2nd circumperp, 1394), (3rd Conway, 1), (extouch, 34488), (3rd extouch, 223), (Garcia-reflection, 1), (2nd Hatzipolakis, 34489), (hexyl, 1), (Hutson intouch, 1), (incentral, 1419), (incircle-circles, 1), (intangents, 1), (intouch, 1), (Lemoine, 34490), (Macbeath, 34491), (medial, 223), (mixtilinear, 1697), (3rd mixtilinear, 1420), (4th mixtilinear, 8830), (6th mixtilinear, 1), (7th mixtilinear, 2124), (orthic, 34492), (2nd Pamfilos-Zhou, 34493), (2nd inner-Soddy, 34494), (2nd outer-Soddy, 34495), (Steiner, 34496), (symmedial, 34497), (Yff contact, 30719)
  The Bevan-antipodal triangle is orthologic to the following triangles with the given centers:

  (ABC, 1, 40), (ABC-X3 reflections, 1, 1), (anti-Aquila, 1, 3), (anti-Ara, 1, 1902), (5th anti-Brocard, 1, 12197), (2nd anti-circumperp-tangential, 1, 3057), (anti-Euler, 1, 6361), (anti-inner-Grebe, 1, 1703), (anti-outer-Grebe, 1, 1702), (anti-Mandart-incircle, 1, 10306), (anticomplementary, 1, 962), (Aquila, 1, 7991), (Ara, 1, 9911), (1st Auriga, 1, 12458), (2nd Auriga, 1, 12459), (5th Brocard, 1, 12497), (2nd circumperp tangential, 1, 22770), (Ehrmann-mid, 1, 22793), (Euler, 1, 10), (3rd extouch, 34498, 31965), (outer-Garcia, 1, 4), (Gossard, 1, 12696), (inner-Grebe, 1, 12697), (outer-Grebe, 1, 12698), (Johnson, 1, 12699), (inner-Johnson, 1, 12700), (outer-Johnson, 1, 5812), (1st Johnson-Yff, 1, 1836), (2nd Johnson-Yff, 1, 12701), (Lucas homothetic, 1, 22841), (Lucas(-1) homothetic, 1, 22842), (Mandart-incircle, 1, 65), (medial, 1, 946), (5th mixtilinear, 1, 7982), (3rd tri-squares-central, 1, 13912), (4th tri-squares-central, 1, 13975), (X3-ABC reflections, 1, 12702), (inner-Yff, 1, 5119), (outer-Yff, 1, 46), (inner-Yff tangents, 1, 12703), (outer-Yff tangents, 1, 12704)

  The Bevan-antipodal triangle is parallelogic to the following triangles with the given centers: (1st Parry, 1, 9811), (2nd Parry, 1, 9810)

  This construction can be generalized for an arbitrary point P (instead of V), and is, in fact, the antipedal triangle of P. Therefore, the Bevan-antipodal triangle is the antipedal triangle of X₄₀. (Randy Hutson, November 17, 2019)

  The Bevan-antipodal triangle is also the anticevian triangle of X₅₇. (Randy Hutson, January 17, 2020)

• X₃₄₅₀₅–X₃₄₅₁₁ — Points of the Moses-Steiner osculatory triangle

  based on notes from Peter Moses, October 19-22, 2019. Suppose that Γ is a curve that circumscribes a triangle ABC and that A', B' C' are the centers of the osculating circles at A, B, C, respectively. The triangle A'B'C' is here named the Moses-Γ osculatory triangle. If Γ = Steiner circumellipse, then A'B'C' is the Moses-Steiner osculatory triangle, for which

  A' = -3 a^4+2 a^2 b^2-b^4+2 a^2 c^2+2 b^2 c^2-c^4 : a^2 (a^2+b^2-c^2) : a^2 (a^2-b^2+c^2)

  B' = b^2 (a^2+b^2-c^2) : -a^4+2 a^2 b^2-3 b^4+2 a^2 c^2+2 b^2 c^2-c^4 : b^2 (-a^2+b^2+c^2)

  C' = c^2 (a^2-b^2+c^2) : c^2 (-a^2+b^2+c^2) : -a^4+2 a^2 b^2-b^4+2 a^2 c^2+2 b^2 c^2-3 c^4,

  and the three osculating circles meet in the Steiner point, X₉₉. If you have GeoGebra, you can view Moses-Steiner osculatory triangle (Moses-Steiner%20osculatory.ggb).

  Let D be the point other than X₉₉ in which the osculating circles and B and C meet, so that D is the reflection of X₉₉ in the A-sideline of the Moses-Steiner osculatory triangle, and D is given by

  D = a^2-b^2-c^2 : -a^2+b^2+2 c^2 : -a^2+2 b^2+c^2,

  Define E and F cyclically. The triangle DEF is here named the Moses-Steiner reflection triangle. See X₃₄₅₁₂-X₃₄₅₁₄.

• X₃₄₅₂₀–X₃₄₅₄₄ — Vu points

  based on notes from Vu Thanh Tung and Vu Quoc My, October 24, 2019. Let P =\(p:q:r\) (barycentrics) be a point in the plane of a triangle ABC. Let A' be the point, other than P, in which the line AP meets the circle (PBC), and define B' and C' cyclically; the triangle A'B'C' is called the circlecevian triangle of P with respect to triangle ABC by Floor van Lamoen ( Hyacinthos # 10039).

  Let T_A and T_A' be the tangents to (PBC) at P and A', respectively.

  Let A₁ = T_A∩BC and A₂ = T_A'∩BC.

  Define B₁ and C₁ cyclically, and define B₂ and C₂ cyclically.

  Then A₁,B₁,C₁ are collinear and A₂,B₂,C₂ are collinear.

  The trilinear pole of the line B₁C₁ is given by Q₁(P) = 1/(-a^2 q r + c^2 q (q + r) + b^2 r (q + r)) : : , here named the 1st Vu point of P.

  The trilinear pole of the line B₂C₂ is given by Q₂(P) = p^2 (-a^2 q r + c^2 q (q + r) + b^2 r (q + r)) : : , here named the 2nd Vu point of P.

  Examples:

  Q₁(X₁) = X₅₇ and Q₂(X₁) = X₉

  Q₁(X₂) = X₅₉₈ and Q₂(X₂) = X₅₉₉

  Q₁(X₃) = X₂ and Q₂(X₃) = X₃₂

  Q₁X(4)) = X₂₀₅₂ and Q₂(X₄) = X₆

  If P lies on the circumcircle, then Q₁(P) is the trilinear pole of the tangent to the circumcircle at P, and Q₂(P) = X₆. (Randy Hutson, October 24, 2019)

  Q₂(P) is the X₂-Ceva conjugate of the Dao image of P. (Randy Hutson, November 17, 2019)

• X₃₄₅₅₁–X₃₄₅₆₂ — Bankoff equilateral triangles

  contributed by César Eliud Lozada, October 29, 2019. Let OA₁A₂, OB₁B₂, OC₁C₂ be three congruent equilateral triangles with the same orientation. Then the midpoints of A₂B₁, B₂C₁ and C₂A₁ are vertices of an equilateral triangle. (Martin Gardner, The Asymmetric Propeller, The College Mathematics Journal, Vol. 30, No. 1, Jan. 1999, pp. 18-22, based on the article by Leon Bankoff, Paul Erdös, and Murray Klamkin, The asymmetric propeller, Mathematics Magazine, 46: 5, 1973, pp 270-272)^{1 2}.

  Application: Let ABC be a triangle with circumcenter O. Centering at O, let A_b be the rotation of A toward B by an angle |π/6| and let A_c be the rotation of A toward C by the same angle |π/6| (triangle OA_bA_c is an equilateral triangle). Build (B_c, B_a) and (C_a, C_b) cyclically and let A_m, B_m, C_m be the midpoints of B_cC_b, C_aA_c, A_bB_a, respectively. Then, by the precedent theorem, A_mB_mC_m is equilateral.

  The triangle A_mB_mC_m will be named here the (ABC)-Bankoff equilateral triangle.

  ¹ The original work by Bankoff, Erdös, and Klamkin refers to three equilateral triangles, not necessarily congruent, but with the same result, i.e, the indicated midpoints are vertices of another equilateral triangle. Martin Gardner added:

  …The original propeller theorem goes back at least to the early 1930’s and is of unknown origin. It concerns three congruent equilateral triangles with corners meeting at a point as shown shaded in Figure 1. The triangles resemble the blades of a propeller.

  …They showed that the three equilateral triangles need not be congruent. They can be of any size, as shown in Figure 2, and the theorem still holds.

  … Later, Bankoff made three further generalizations. As far as I know they have not been published.

  Second generalization: The propeller triangles need not meet at a point. They may meet at the corners of any equilateral triangle, as shown in Figure 3. Third generalization: The propeller triangles need not be equilateral! They need only be similar triangles of any sizes that meet at a point. The midpoints of the three added lines will then form a triangle similar to each of the propellers, as shown in Figure 4. Fourth generalization: The similar triangles need not meet at a point! If the propellers meet at the corners of a fourth triangle of any size, provided it is similar to each propeller, the midpoints of the added lines will form a triangle similar to each propeller. Vertices of the interior triangle must touch corresponding corners of the propellers.

  See the above mentioned figures here (./Bankoff_theorem.gif).

  ² The cited Martin Gardner’s article was referenced by Richard Guy and Antreas Hatzipolakis in Hyacinthos #26 & #28 (December 28, 1999).

• X₃₄₅₆₅–X₃₄₅₇₄ — Centers related to the Moses-Jerabek and Moses-Lemoine conics

  contributed by Randy Hutson, October 31, 2019. The Moses-Jerabek and Moses-Lemoine conics are introduced in the preamble before X₃₄₄₂₆.

• X₃₄₅₇₈–X₃₄₅₈₁ — Cyclologic centers

  contributed by Vu Thanh Tung and Vu Quoc My, October 31, 2019. Let P =\(p:q:r\) (barycentrics) be a point in the plane of a triangle ABC, and let A' be the point, other than P, where the line AP meets the circle (PBC). Let U =\(u:v:w\) be a point (as a function of a, b, c), and let A″ = U-of-A'BC. Define B″ and C″ cyclically. Let T* = A″B″C″. The triangles ABC and T* are cyclologic; i.e., the circumcircles of A″BC, AB″C, ABC″ concur in a single point, called the ABC-to-T* cyclologic center, denoted by S(P,U).

• X₃₄₅₈₅–X₃₄₅₉₃ — Points related to the circumellipse of the medial and incentral triangles

  based on notes from Dasari Naga Vijay Krishna and Peter Moses, November, 2019. The circumellipse of the medial and incentral triangles (the CEMIT), given by the equation

  b*c*x^2 - a*c*x*y - b*c*x*y + a*c*y^2 - a*b*x*z - b*c*x*z - a*b*y*z - a*c*y*z + a*b*z^2 = 0,

  has center X₁₁₂₅ and perpsector X₃₄₅₈₅. The axes are parallel to the asymptotes of the Feuerbach hyperbola, and

  Major axis has length = (b + c) (c + a) (a + b) (R + |OI|) / (32 R s), in the line X₁₁₂₅X₃₃₀₇

  Minor axis has length = (b + c) (c + a) (a + b) (R - |OI|) / (32 R s), in the line X₁₁₂₅X₃₃₀₈
  The CEMIT passes through the following points:

  vertices of the medial triangle

  vertices of the incentral triangle

  vertices of the anti-Aquila triangle

  X_i for these i: 11, 214, 244, 1015, 8054, 8299, 10494, 14714, 17417, 17419, 17421, 17761, 17793, 34586, 34587, 34588, 34589, 34590, 34591, 34592, 34593.

  If P = p:q;r lies on the circumcircle then the point \(f(P) = a (q c (a + c) + r b (a + b))\): : lies on the CEMIT. Inversely, if U lies on the CEMIT, then the point \(g(U) = (a (a + b) (a + c) (-u + v + w)\): : lies on the circumcircle.

  The appearance of (i,j) in the following list means that f(X_i) = X_j: (99,1015), (100,244), (101,17761), (104,34586), (105,8299), (106,34587), (109,34589), (110,11), (741,17793), (759,214), (901,34590), (934,345191), (1113,34592), (1114,34593), (34594,8054).

  The CEMIT is the bicevian conic of X₁ and X₂. For a discussion of general bicevian conics, see Bernard Gibert, Bicevian Conics and CPCC Cubics

• X₃₄₆₀₃–X₃₄₇₅₂ — HR-ellipses

  contributed by César Eliud Lozada, November 7, 2019. Let T'=A'B'C' and T″=A″B″C″ be two homothetic triangles. Denote (B'_a, C'_a) the reflections of B' and C' in A″, respectively, and define (C'_b, A'_b) and (A'_c, B'_c) cyclically. Then these six points lie on an ellipse here named the HR-ellipse of T’ to T” (letters H and R stands for homothetic and reflections). By swapping T' and T″, the HR-ellipse T″ to T' is found.

  Suppose T' and T″ are homothetic to the reference triangle ABC, with H' = homothetic center (ABC, T') and H″=homothetic center (ABC, T″). Denote λ’=AA'/AH' and λ”=AA″/AH″. Then, if H', H″ have normalized barycentrics coordinates H' =\(x' : y' : z'\) and H″ = x” : y” : z”, the centers O', O″ of the HR-ellipses T' to T″ and T″ to T' are:

  O' = (3*x’ - 1)*λ’ - 2*(3*x” - 1)*λ” - 1 : :

  O″ = (3*x” - 1)*λ” - 2*(3*x’ - 1)*λ’ - 1 : :

  In general, { H', H″, O' , O″} are aligned, then O' = complement-of-O″ with respect to T' and O″ = complement-of-O' with respect to T″.

• X₃₄₇₇₃–X₃₄₇₉₅ — Ellipsologic centers

  contributed by César Eliud Lozada, November 14, 2019. Let T'=A'B'C' and T″=A″B″C″ be two triangles. Denote as E'_a the ellipse passing through A' and having foci B″ and C″. Define E'_b and E'_c cyclically. If theses three ellipses intersect in a unique point Q', then Q' is here named the ellipsologic center of T’ to T”.

  Note: The existence of the ellipsologic center of T' to T″ does not imply the existence of the ellipsologic center of T″ to T'.

  It is known that “if three conics are such that each pair has one common focus and two and only two real points of intersection, the three chords of visible intersection are concurrent” (E. Neville, “A focus-sharing set of three conics”, Mathematical Gazette 20(239) (1936) 182-183, and B. Lawrence, “Note on focus sharing conics”, Mathematical Gazette 21(243) (1937) 160-161.

  The point of intersection, Q_i, of the common chords mentioned above does not lie necessarily on any of the conics, but if it lies on one of the conics then it is obviously the intersection of the three conics. Lawrence gives the following interesting alternative construction of Q_i:

  Assume the major axes of the ellipses E'_a, E'_b, E'_c are λ_a = A'B″+A'C″, λ_b = B'C″+B'A″ and λ_c = C'A″+C'B″ and suppose λ_c is the longest axis. If λ’_a = λ_c - λ_b ≥ 0 and λ’_b = λ_c - λ_a ≥ 0 then Q_i is the point at distances λ_a*λ’_a/(2*B″C″) and λ_b*λ’_b/(2*C″A″) from the perpendicular bisectors of B″C″ and C″A″, respectively.

  For definitions of these triangles see here (IndexOfTrianglesReferencedInETC.html). All pairs of triangles in this index were compared and results are showed in the above list.

• X₃₄₈₀₇–X₃₄₈₂₁ — Points associated with perspeconics

  contributed by Clark Kimberling and Peter Moses, November 18, 2019. Suppose that A'B'C' and A″B″C″ are perspective central triangles, so that the points B'C'∩A″B″, B'C'∩A″C″, C'A'∩B″C″, C'A'∩B″A″, A'B'∩C″B″, A'B'∩C″A″ are distinct and lie on a conic. In the preamble just before X₁₅₂₅₄ the conic is named the perspeconic of A’B’C’ and A’’B’’C’’.

  The perspeconic of ABC and the circumcevian triangle of a point P =\(p : q : r\) (barycentrics) is given by

  a^2 q^2 r^2 (c^2 q+b^2 r) x^2-p q r (2 b^2 c^2 p^2+a^2 c^2 p q+a^2 b^2 p r+a^4 q r) y z + (cyclic) = 0,

  center = p (2 b^2 c^2 p q+2 a^2 c^2 q^2+2 b^2 c^2 p r-a^4 q r+a^2 b^2 q r+a^2 c^2 q r+2 a^2 b^2 r^2) : :

  perspector = p (2 b^2 c^2 p+2 a^2 c^2 q+a^2 b^2 r) (2 b^2 c^2 p+a^2 c^2 q+2 a^2 b^2 r) : :

  If P lies inside ABC, the conic is an ellipse; otherwise, a hyperbola. As a degenerate case, if P lies on the circumcircle, then the conic is the line tangent to the circumcircle at P. If P is on the line X₁₈₇X₂₃₇, the conic consists of two lines that pass through X₆.

• X₃₄₈₂₂–X₃₄₈₅₂ — Centers of perspeconics of cevian and anticevian triangles

  contributed by Clark Kimberling and Peter Moses, November 20, 2019. Suppose that A'B'C' and A″B″C″ are perspective central triangles, so that the points B'C'∩A″B″, B'C'∩A″C″, C'A'∩B″C″, C'A'∩B″A″, A'B'∩C″B″, A'B'∩C″A″ are distinct and lie on a conic. In the preamble just before X₁₅₂₅₄ the conic is named the perspeconic of A’B’C’ and A’’B’’C’’. Suppose that P =\(p : q : r\) and U =\(u : v : w\). The cevian triangle of P is perspective to the anticevian triangle of U, the perspector being, by definition, the P-Ceva conjugate of U. The perspeconic of the two triangles is given by

  q^2 r^2 ((q r u + p r v - p q w) (q r u - p r v + p q w) x^2 + 2 p^2 u (q r u + p r v + p q w) y z) + (cyclic) = 0,

• X₃₄₈₆₄–X₃₄₈₈₉ — Circumcevian-inversion perspectors

  based on notes by Suren, (Circumcevian Inversion Perspector (Circumcevian%20Inversion%20Perspector.pdf)), contributed by Clark Kimberling and Peter Moses, November 20, 2019. Suppose that P =\(p : q : r\) (barycentrics) is a point in the plane of a triangle ABC. Let A*B*C* be the circumcevian triangle of P, and let A' be the inverse-in-circumcircle of the reflection of P in A*; define B' and C' cyclically. The triangle A'B'C', here named the circumcevian-inversion triangle of P, is perspective to ABC in the point

  P' = p U + \(a^{2}\) (\(b^{2}\) + \(c^{2}\) - \(a^{2})\) V : : , where U = \(a^{2}\) \(b^{2}\) \(c^{2}\) (p+q+r) and V = \(a^{2}\) q r + \(b^{2}\) r p + \(c^{2}\) p q.

  If P lies on the circumcircle, then P' = P; if P lies on the line at infinity, then P' = X₃.

  If P lies on the Stammler circle, then P' lies on the circumcircle.

  If P on the Euler line is given by P = g*X₂ + h*X₃, then P' = 9 a^2 b^2 c^2 g (g + h) X₂ + (9 a^2 b^2 c^2 h (3 g + 2 h) + 4 (a^2 + b^2 + c^2) g^2 S^2) X₃ .

  If P on the line X₁X₃ is given by P = g*X₁ + h*X₃, then P' = g (g + h) R s X₁ + (h (3 g + 2 h) R s + g^2 S) X₃.

  The appearance of (i,j) in the following list means that the circumcevian-inversion perspector of X_i is X_j:

  See the preamble for X₃₅₀₀₀-X₃₅₀₀₂.

• X₃₄₈₉₂–X₃₄₉₀₂ — Vu circlecevian points

  contributed by Vu Thanh Tung and Vu Quoc My, November 20, 2019, with extensions and editing by Clark Kimberling and Peter Moses, November 22, 2019. Let P be a point in the plane of a triangle ABC, and let A' be the point, other than P, in which the line AP meets the circle (PBC). Define B' and C' cyclically, so that A'B'C' is the circlecevian triangle of P, as in the preamble just before X₃₄₂₀.

  Now let A'B'C' and A″B″C″ be, respectively, the circlecevian triangles of P =\(p : q : r\) and U =\(u : v : w\) (barycentrics). Let A₁ = BC∩A'A″, and define B₁ and C₁ cyclically. Then AA₁, BB₁, CC₁ concur in a point V(P,U), here named the Vu circlecevian point of P and U, given by

  V(P,U) = p*u / (p*(p+q+r)*(a^2*v*w + b^2*w*u + c^2*u*v) - u*(u+v+w)*(a^2*q*r + b^2*r*p + c^2*p*q)) : : .

  The appearance of (i,j,k) in the following list means that V(X_i,X_j) = X_k:

  (1,2,34892), (1,3,1807), (1,4,80), (1,6,34893), (1,7,3254), (1,8,12641), (1,9,34894), (1,10,34895), (1,11,34896), (2,3,34897), (2,4,671), (2,6,34898), (2,10,34849), (3,5,34900), (3,4,265), (3,5,34900), (3,6,895), (3,8,34901), (3,4,902), (4,5,1263), (4,6,316), (4,7,1156), (4,8,1320), (4,9,3245), (4,10,11599), (13,14,2), (15,16,323), (485,486,8781)

  The Vu circlecevian point of P and U lies on the conic {{A, B, C,P,U}}. (Randy Hutson, November 26, 2019)

  Given three points P,Q,R, then 6 points A, B, C, V(P,Q), V(Q,R), V(R,P) lie on a conic. (Tran Quang Hung, Euclid #225)

• X₃₄₉₁₄–X₃₄₉₂₁ — Images of Vu T-transform

  contributed by Vu Thanh Tung and Vu Quoc My, November 23, 2019, with extensions and editing by Clark Kimberling and Peter Moses, November 23, 2019. Let P be a point in the plane of a triangle ABC, and let A' be the point, other than P, in which the line AP meets the circle (PBC). Define B' and C' cyclically. The triangle A'B'C' is called the circlecevian triangle of P with respect to triangle ABC by Floor van Lamoen ( Hyacinthos # 10039).

  Let I be the incenter of triangle ABC, and let A0 = BC∩IA'; define B0 and C0 cyclically. Then AA0, BB0, CC0 concur in a point, T(P), here named the Vu T-transform of P. Barycentrics are given by

  T(P) = p / (b*c*p^2 + b*c*p* q + c^2*p*q + b^2*p*r + b*c*p*r + a^2*q*r) .

  The appearance of (i,j) in the following list means that T(X_i) = X_j:

  (1,1), (2,34914), (3,7100), (4,79), (5,34915), (6,34916), (7,34917), (8,34918), (9,34919), (10,34920)

• X₃₄₉₂₄–X₃₄₉₃₅ — Orthology centers associated with osculatory triangle of K721

  contributed by Peter Moses, November 24, 2019. Suppose that a curve K passes through A, B, C and has osculating circle O_a at A. Let A' be the center of O_a, and define B' and C' cyclically. The triangle A'B'C' is here named the osculatory triangle of K.

• X₃₄₉₃₆–X₃₄₉₉₁ — Polar reciprocal conics

  contributed by César Eliud Lozada, November 24, 2019. Let K₁, K₂ be two conics (circles included) with centers O₁ and O₂, respectively. The locus of the poles with respect to K₁ of the tangents to K₂ is another conic Φ₂₁ named the reciprocal polar conic of K₂ with respect to K₁. By swapping K₁ and K₂, the reciprocal polar conic Φ₁₂ of K₂ with respect to K₁ is found.

  The center of Φ₂₁ is the pole, with respect to K₁, of the polar of O₁ wrt K₂ and, similarly, the center of Φ₁₂ is the pole, with respect to K₂, of the polar of O₂ wrt K₁. (Reference: Eagles, T.H.: Constructive Geometry of Plane Curves, McMillan and Co, London, 1885. pp 201-202.)

• X₃₄₉₉₂–X₃₄₉₉₉ — Perspectors involving circlecevian triangles

  contributed by Clark Kimberling and Peter Moses, November 24, 2019. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC, and let A' be the point, other than P, in which the line AP meets the circle (PBC). Define B' and C' cyclically. The triangle A'B'C' is called the circlecevian triangle of P with respect to triangle ABC by Floor van Lamoen ( Hyacinthos # 10039); see the preambles just before X₃₄₈₉₂ and X₃₄₉₁₄. The A-vertex of the circlecevian triangle of P is given by A' = -a^2 q r (p + q + r) : q (c^2 p q + b^2 p r + a^2 q r) : r (c^2 p q + b^2 p r + a^2 q r).

  Examples:

  If P = X₁, then A'B'C' = excentral triangle.

  If P = X₄, then A' = reflection of A in BC, and B' and C' are defined cyclically.

  If P = X₂, then A' = -3 a^2 : a^2 + b^2 + c^2 : a^2 + b^2 + c^2.

  If P = X₃, then A' = (a^2 + b^2 - c^2)(a^2 - b^2 + c^2) : b^2 (-a^2 + b^2 - c^2), -c^2 (a^2 + b^2 - c^2).

  If P = X₆, then A' = a^2 + b^2 + c^2 : -3 b^2 : -3 c^2.

  If P = X₃₁, then A' = -b*c*(a^3 + b^3 + c^3) : b^3*(a*b + a*c + b*c), c^3*(a*b + a*c + b*c).

  If P = X₇₅, then A' = -a^2*(a*b + a*c + b*c) : c*(a^3 + b^3 + c^3) : b*(a^3 + b^3 + c^3).

  If P = X₇₆, then A' = -a^2*(a^2*b^2 + a^2*c^2 + b^2*c^2) : c^2*(a^4 + b^4 + c^4) : b^2*(a^4 + b^4 + c^4).

  Note from Vu Thanh Tung, November 25, 2019: the circlecevian triangle of an arbitrary point P =\(p : q : r\) is perspective to the excentral triangle. The perspector, P', is given by

  P' = a ( a^3 q^2 r^2 - b^3 p^2 r^2 -c^3 p^2 q^2 + a b c p^2 q r + a b c p q^2 r + a b c p q r^2 - b^2 c p^2 q r - b c^2 p^2 q r - a^2 b p q r^2 - a^2 c p q^2 r + a c^2 p q^2 r + a b^2 p q r^2 ) : :

• X₃₅₀₀₀–X₃₅₀₀₂ — Secondary pre-circumcevian-inversion points

  contributed by Peter Moses, November 25, 2019. Suppose that a point X is the circumcevian-inversion perspector of a point P =\(p : q : r\), as in the preamble just before X₃₄₈₆₄. There is a second point, Q, of which X is the circumcevian-inversion perspector of Q. The point Q, here named the secondary pre-circumcevian-inversion point of P, is given by

  Q = a^2 (b^2 c^2 (a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) p^2+c^2 (a^6+a^4 b^2+a^2 b^4-3 b^6-3 a^4 c^2+5 b^4 c^2+3 a^2 c^4-b^2 c^4-c^6) p q+3 a^2 b^2 c^2 (a^2-b^2-c^2) q^2+b^2 (a^6-3 a^4 b^2+3 a^2 b^4-b^6+a^4 c^2-b^4 c^2+a^2 c^4+5 b^2 c^4-3 c^6) p r+a^2 (a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2+4 b^2 c^2+c^4) q r+3 a^2 b^2 c^2 (a^2-b^2-c^2) r^2) : : .

  Let Cip(P) denote the circumcevian-inversion perspector of a point P.

  Example 1: Cip(X₁) = X₃₅, the circumcevian-inversion perspector of X₁; also, Cip(X₃₅₀₀₀) = X₃₅, so that X₃₅₀₀₀ is the secondary pre-circumcevian-inversion point of X₁.

  Example 2: Cip(X₂) = X₇₄₉₆, the circumcevian-inversion perspector of X₂; also, Cip(X₃₅₀₀₁) = X₇₄₉₆, so that X₃₅₀₀₁ is the secondary pre-circumcevian-inversion point of X₂.

  Example 3: Cip(X₄) = X₃₅₂₀, the circumcevian-inversion perspector of X₄; also, Cip(X₁₈₈₅₉) = X₃₅₂₀, so that X₁₈₈₅₉ is the secondary pre-circumcevian-inversion point of X₄.

  Example 4: Cip(X₆) = X₅₇₄, the circumcevian-inversion perspector of X₆; also, Cip(X₃₅₀₀₂) = X₅₇₄, so that X₃₅₀₀₂ is the secondary pre-circumcevian-inversion point of X₆.

  Example 5: Cip(X₂₀) = X₇₄₈₈, the circumcevian-inversion perspector of X₂₀; also, Cip(X₂₀₇₀) = X₇₈₈, so that X₂₀₇₀ is the secondary pre-circumcevian-inversion point of X₂₀.

• X₃₅₀₀₅–X₃₅₀₀₉ — Centers related to Vu circlecevian points

  contributed by Randy Hutson, November 26, 2019. Vu circlecevian points are introduced in the preamble before X₃₄₈₉₂.

• X₃₅₀₁₂–X₃₅₀₁₅ — Centers on the Sherman line

  :contributed by Peter Moses, November 29, 2019. See Paul Yiu, Sherman’s Fourth Side of a Triangle .

• X₃₅₀₂₅–X₃₅₀₄₈ — Points on the permutation ellipse of X₁

  contributed by Clark Kimberling and Peter Moses, December 1, 2019. Suppose that P =\(p : q : r\) (barycentrics) is a point other than X₂ =\(1 : 1 : 1\) in the plane of a triangle ABC. Let T denote the triangle with vertices

  \(p : q : r\)

  \(q : r : p\)

  \(r : p : q\)

  Let T' denote the obverse of T, defined in the preamble just before X₂₄₃₀₇ by vertices

  \(p : r : q\)

  \(q : p : r\)

  \(r : q : p\)

  The six points, corresponding to the permutations pqr, qrp, rpq, prq, qpr, rqp, lie on the permutation ellipse of P, as defined in the preamble just before X₃₄₃₄₁, given by

  (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  The center and perspector of E(P) are both the centroid, X₂. Let u = length of semi-major axis of E(P) and v = length of semi-minor axis of E(P). Then

  u^2 = 1/(9 (p + q + r)^2) (p^2 + q^2 + r^2 - q r - r p - p q)((a^2 + b^2 + c^2) + \(\sqrt{((a^{2} + b^{2} + c^{2})^{2} - 12 S^{2})}\))

  v^2 = 1/(9 (p + q + r)^2) (p^2 + q^2 + r^2 - q r - r p - p q)((a^2 + b^2 + c^2) - \(\sqrt{((a^{2} + b^{2} + c^{2})^{2} - 12 S^{2})}\))

  radius of orthoptic circle = (2 (a^2 + b^2 + c^2) (p^2 + q^2 + r^2 - q r - r p - p q))/(9 ( p + q + r)^2)

  If P' lies on E(P), then E(P') = E(P). Moreover, if U =\(u : v : w\) is a point, other than P, then the point, other than P', in which the line UP' meets E(P), is here named the E(P,U)-antipode of P’, denoted by E(P,P',U) and given by

  E(P,P',U) =\(f(p,q,r,u,v,w) : f(q,r,p,v,w,u) : f(r,p,q,u,v,w)\), where

  \(f(p,q,r,u,v,w) = (q^{3} + q^{2} r + q r^{2} + r^{3} - p q r) u^{2} + p (q r + r p + p q) (v^{2} + w^{2})\)

  - p (p^2 + q^2 + r^2)v w + (p (p q - 2 r^2 - 2 q r) + q (q^2 - r^2)) w u + (p (p r - 2 q^2 - 2 q r) + r (r^2 - q^2)) u v .

• X₃₅₀₆₆–X₃₅₀₉₅ — Points on the Steiner inellipse (the permutation ellipse E(X₁₁₅)

  contributed by Clark Kimberling and Peter Moses, December 4, 2019. This section extends from preambles just before X₃₄₃₄₁ and X₃₅₀₂₅. The points X₃₅₀₆₆-X₃₅₀₉₅ are of the form E(X₁₁₅,X_k,X₁₁₅), where E(X₁₁₅) is the Steiner inellipse.

• X₃₅₁₀₁–X₃₅₁₀₄ — 4th intersection of named circumconics

  contributed by César Eliud Lozada, December 4, 2019. If K₁, K₂ are two circumconics of ABC then the 4th intersection of K₁ and K₂ (i.e., the intersection other than A, B, C) is the trilinear pole wrt ABC of the line joining the perspectors of K₁ and K₂. Following is a list of the 4th points of intersection of some named conics:

  X₄ = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and Kiepert hyperbola

  X₁₁₀ = intersection, other than A, B, C, of these conics: Johnson circumconic and MacBeath circumconic

  X₂₆₅ = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and Johnson circumconic

  X₂₉₀ = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and Steiner circumellipse

  X₆₄₈ = intersection, other than A, B, C, of these conics: MacBeath circumconic and Steiner circumellipse

  X₆₇₁ = intersection, other than A, B, C, of these conics: Kiepert hyperbola and Steiner circumellipse

  X₈₉₅ = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and MacBeath circumconic

  X₉₀₃ = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and Steiner circumellipse

  X₁₂₄₆ = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and Jerabek hyperbola.

  X₂₄₈₁ = intersection, other than A, B, C, of these conics: Feuerbach hyperbola and Steiner circumellipse

  X₂₉₈₆ = intersection, other than A, B, C, of these conics: Kiepert hyperbola and MacBeath circumconic

  X₂₉₈₉ = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and MacBeath circumconic

  X₆₅₂₈ = intersection, other than A, B, C, of these conics: Johnson circumconic and Steiner circumellipse

  X₈₇₅₉ = intersection, other than A, B, C, of these conics: Feuerbach hyperbola and MacBeath circumconic

  X₃₅₀₉₆ = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and Johnson circumconic

  X₃₅₀₉₇ = intersection, other than A, B, C, of these conics: Feuerbach hyperbola and Johnson circumconic

  X₃₅₀₉₈ = intersection, other than A, B, C, of these conics: Johnson circumconic and Kiepert hyperbola

• X₃₅₁₁₀–X₃₅₁₃₅ — Barycentric positive square roots of points on the Steiner inellipse

  contributed by Clark Kimberling and Peter Moses, December 4, 2019. This section extends from the introduction to permutation ellipses in the preamble just before X₃₅₀₂₅. The points X₃₅₁₀₁-X₃₅₁₀₄ are positive square roots of points the form E(X₁₁₅,X_k,X₁₁₅), where E(X₁₁₅) is the Steiner inellipse.

  Suppose that a point \(f : g : h\) lies on the line at infinity (i.e., f + g + h = 0). Then the point \(f^{2} : g^{2} : h^{2}\) lies on the Steiner inellipse, E(X₁₁₅). Conversely, if \(F^{2} : G^{2} : H^{2}\) lie on the ellipse, then the following six points lie on the line at infinity:

  \(f : g : h\) \(g : h : f\) \(h : f : g\) \(f : h : g\) \(g : f : h\) \(h : g : f\)

  where f, g, h are the positive square roots of F,G,H.

  The isogonal conjugates of the points X₃₅₁₀₁-X₃₅₁₀₄, all on the circumcircle, are indexed as X₃₅₁₀₅-X₃₅₁₀₈.

• X₃₅₁₃₆–X₃₅₁₈₀ — Points on the Steiner circumellipse

  contributed by Clark Kimberling and Peter Moses, December 5, 2019. The Steiner circumellipse, E(X₉₉), is the anticomplement of the Steiner inellipse, E(X₁₁₅), and the points in this section are presented as anticomplements of points on the Steiner inellipse. As described in the preamble just before X₃₄₃₄₁, the two ellipses are permutation ellipses; that is, if \(p : q : r\) is on the ellipse then all six of the permutation points, abbreviated by pqr,qrp,rpq,prq,qpr,rqp, are also on the ellipse.

• X₃₅₁₈₂–X₃₅₁₉₁ — Haedus transforms

  contributed by César Eliud Lozada, December 5, 2019. Let ABC be an acute triangle with circumcircle ω. Let t be a tangent line to ω and denote \(t_{a}\), \(t_{b}\), \(t_{c}\) the lines obtained by reflecting t in the lines BC, CA and AB, respectively. Then the circumcircle ω′ of the triangle determined by the lines \(t_{a}\), \(t_{b}\), \(t_{c}\) is tangent to the circle ω. References: 2011 IMO Shortlist :g:`G_{8} <G_8>` (JPN) problem 6, IMOgeometry and AOPS.

  If P =\(u:v:w\) (trilinears) is the touchpoint of t and ω and H(P) is the touchpoint of ω and ω′, then

  \(H(P) = u S_{A} ((a^{2} u^{2} + 3 b^{2} v^{2} + 3 c^{2} w^{2}) S_{A}^{2} S_{B} S_{C} a^{3} u\)

  + ((2*SA - SB)*S^2 - 3*(SB + SC)*SA^2)*SA*a^4*b*u^2*v

  + ((2*SA - SC)*S^2 - 3*(SB + SC)*SA^2)*SA*a^4*c*u^2*w

  + (S^2 - 3*SA*SB)*SB*SC^2*b^2*c*v^2*w

  + (S^2 - 3*SA*SC)*SC*SB^2*c^2*b*w^2*v

  - (S^2 + SA*SC)*SB^2*SC*b^3*v^3

  - (S^2 + SA*SB)*SC^2*SB*c^3*w^3

  + (5*S^2 + 6*SA^2 - 24*R^2*SA - 4*SB*SC)*S^2*a^3*b*c*u*v*w) : :
  \(H(P) = IsotomicConjugate( PolarConjugate( BarycentricProduct( P\), X₄-antipode-of-P) ) )

  where X₄-antipode-of-P means the intersection, other than P, of ω and the line X₄P, as defined in the preamble just before X₂₆₇₀₀. The point H(P) is here named here the Haedus transform of P.

• X₃₅₁₉₂–X₃₅₁₉₆ — Points associated with the cubic pK(X(35192,X₂₁)

  contributed by Clark Kimberling and Peter Moses, December 8, 2019. Suppose that A'B'C' is the circumcevian-inversion triangle of X₁, as defined in the preamble just before X₃₄₈₆₄. The locus of a point Q such that the cevian triangle of Q is perspective to A'B'C' is the cubic pK(X₃₅₁₉₂,X₂₁), which passes through A, B, C, and X_i for these i: 1, 3, 21, 35, 3467, 11107, 35193, 35194, 35195, 35196.

• X₃₅₁₉₇–X₃₅₂₀₁ — Points associated with the cubic pK(X₅₀,X₁)

  contributed by Clark Kimberling and Peter Moses, December 8, 2019. Suppose that A'B'C' is the circumcevian-inversion triangle of X₁, as defined in the preamble just before X₃₄₈₆₄. The locus of a point Q such that the anticevian triangle of Q is perspective to A'B'C' is the cubic pK(X₅₀,X₁), which passes through A, B, C, the vertices of the incentral triangle, and X_i for these i: 1, 35, 36, 1094, 1095, 2169, 5353, 5357, 6126, 6149, 7343, 35197, 35198, 35199, 35200, 35201.

• X₃₅₂₀₂–X₃₅₂₁₀ — Points associated with the circumcevian-inversion triangle of X₁

  contributed by Clark Kimberling and Peter Moses, December 8, 2019. Let T = circumcevian-inversion triangle of X₁, denoted by A'B'C' in the preamble just before X₃₄₈₆₄. The triangle T is perspective to each triangle in Column 2 of the following table. These triangles are described in César Lozada’s Index of Triangles Referenced in ETC (IndexOfTrianglesReferencedInETC.html). For points associated with the circumcevian-inversion triangle of X₁, see X₃₅₂₃₇-X₃₅₂₅₇.

• X₃₅₂₁₁–X₃₅₂₂₆ — Perspectors of tangential triangle and other triangles

  :contributed by Clark Kimberling and Peter Moses, December 10, 2019.

• X₃₅₂₃₇–X₃₅₂₅₇ — Points associated with the circumcevian-inversion triangle of X₃

  :contributed by Clark Kimberling and Peter Moses, December 11, 2019. Let T = circumcevian-inversion triangle of X₁, denoted by A'B'C' in the preamble just before X₃₄₈₆₄. The triangle T is perspective to each triangle in Column 2 of the following table. These triangles are described in César Lozada’s Index of Triangles Referenced in ETC (IndexOfTrianglesReferencedInETC.html). For points associated with the circumcevian-inversion triangle of X₁, see X₃₅₂₀₂-X₃₅₂₁₀.

• X₃₅₂₅₈–X₃₅₃₀₆ — Centroids of three points on the circumcircle

  :contributed by Peter Moses, December 11, 2019. According to a theorem by Bernard Gibert, every cubic of the type nK0(X6,R), where R =\(u : v : w\), meets the circumcircle in three points, other than A, B, C, and the centroid of those three points is given by

  (-\(3a^{2}\) + \(b^{2}\) + \(c^{2})\)u + \(2a^{2}\)v + \(2a^{2}\)w : :

  For centers X₃₅₂₃₈-X₃₅₃₀₆, the type is written as nK0(X₆,X_k), and the circumcircle as Γ .

  See Table 4. nK0(X6,R).

• X₃₅₃₀₇–X₃₅₃₆₈ — Areal centers

  contributed by César Eliud Lozada, December 12, 2019. Let A'B'C', A″B″C″ be two triangles inscribed in ABC. The areal center of the two triangles is a point S such that the triangles SA’A″, SB’B″, SC’C″ have the same area. (Reference: The Triangles Web)

  Construction of the areal center of two inscribed triangles

  Let A'B'C', A″B″C″ be the two inscribed triangles, L_a the line through A parallel to the line joining the midpoints of B'C″ and B″C'. Construct analogously L_b, L_c. The lines L_a, L_b, L_c intersect at the areal center S.

  (Reference: The Triangles Web)

• X₃₅₃₇₄–X₃₅₄₄₀ — Centers of circles through couples of bicentric pairs

  contributed by César Eliud Lozada, December 14, 2019. The appearance of (i, j, n) in the following list means that the bicentric pairs PU(i) and PU(j) are concyclic on a circle with center X_n:

  (List omitted here.)

• X₃₅₄₄₇–X₃₅₄₆₅ — Source-points for a pair of circumcevian-inversion points

  contributed by Peter Moses, December 16, 2019. The secondary pre-circumcevian-inversion point of a point P is defined just before X₃₅₀₀₀. The [primary] point is defined just before X₃₄₈₆₄. For a list of points P and associated primary and secondary points, see the preamble just before X₃₅₀₀₀.

• X₃₅₅₁₆–X₃₅₅₇₅ — Points on De Longchamps line

  contributed by César Eliud Lozada, December 19, 2019. The De Longchamps line X₃₂₅X₅₂₃ is the isotomic conjugate of the circumcircle. The appearance of (i, j) in the following list means that the isotomic conjugate of X_i is X_j, where X_i is on the circumcircle and X_j is on the De Longchamps line: (list omitted here.)

• X₃₅₅₇₉–X₃₅₅₉₄ — Intersections of Simson lines on the nine-point circle

  contributed by César Eliud Lozada, December 22, 2019. Let :g:`P <P>` and :g:`Q <Q>` be antipodal points on the circumcircle of :g:`ABC << A B C >>`. Then the Simson lines of :g:`P <P>` and :g:`Q <Q>` intersect at a point on the nine-point circle of :g:`ABC << A B C >>`. If X is such point of intersection, and \(P^{-1}\), \(Q^{-1}\) are the isogonal conjugates of P and Q, then:

  X is crosssum of P and Q; therefore, X is the crosspoint of \(P^{-1}\) and \(Q^{-1}\). X is crossdifference of every pair of points on the polar trilinear of \(X^{-1}\). X is center of the rectangular circum-hyperbola {{A, B, C, X₄, \(P^{-1}\), \(Q^{-1}\)}}.

  Moreover, if Y is the intersection, other than A, B, C, of the circum-parabolas {{A, B, C, \(P^{-1}\)}} and {{A, B, C, \(Q^{-1}\)}}, then:

  X is the complement of Y. X is the X₄-Ceva conjugate-of \(Y^{-1}\).

  The appearance of (i, j, k) in the following list means that X_k is the intersection of Simson lines of the circumcircle-antipodal centers X_i and X_j: (list omitted here)

• X₃₅₆₁₀–X₃₅₉₀₀ — Miscellaneous triangles and related centers

  contributed by César Eliud Lozada, December 29, 2019. The following triangles are defined in ETC:

  (ABC)-Bankoff equilateral triangle (shortened to Bankoff-equilateral triangle), in the preamble just before X₃₄₅₅₁ Bevan antipodal, in the preamble just before X₃₄₄₈₈ inverse-in-Conway, in X₁₀₄₃₉ inverse-in-excircles, in X₈₀₅₅ 1st Kenmotu-free-vertices, in X₁₂₃₇₅ 2nd Kenmotu-free-vertices, in X₁₂₃₇₆ Moses-Steiner osculatory, in the preamble just before X₃₄₅₀₄ Moses-Steiner reflection, in the preamble just before X₃₄₅₀₄ orthic axes, in X₂₅₀₁ Yiu tangents, in X₇₄₉₅

  A complete list of centers related to these triangles can be seen here (Attach_X35610.pdf).

• X₃₅₉₅₆–X₃₅₉₆₂ — Points on the permutation ellipse of X₇₅

  contributed by Clark Kimberling and Peter Moses, January 2, 2020. Suppose that P is a point not in {X₂,A, B, C}. The permutation ellipse of P is denoted by E(P). For details, see the preamble to X₃₅₀₂₅-X₃₅₀₄₈.

• X₃₅₉₆₃–X₃₅₉₆₆ — Points on the permutation ellipse of X₆

  contributed by Clark Kimberling and Peter Moses, January 2, 2020. Suppose that P is a point not in {X₂,A, B, C}. The permutation ellipse of P is denoted by E(P). For details, see the preamble to X₃₅₀₂₅-X₃₅₀₄₈.

Part 19

• X₃₆₀₃₃–X₃₆₁₅₁ — Centers associated with trilinear products of circumcircle-P-antipodes

  contributed by Randy Hutson, January 3, 2020. Let P =\(p : q : r\) (trilinears). Then the locus of the trilinear product of circumcircle-P-antipodes is the circumconic with perspector the trilinear product X₆*P = a p : b q : c r. Let L be a line. The trilinear product of the (real or nonreal) circumcircle intercepts of L is the trilinear pole of the X₂-isoconjugate of the isogonal conjugate of L (or equivalently, X₆*L). These intercepts are also circumcircle-P-antipodes for all P on L.

• X₃₆₂₁₅–X₃₆₂₄₀ — Points on permutation ellipses

  contributed by Clark Kimberling and Peter Moses, January 7, 2020. Suppose that P =\(p : q : r\) (barycentrics) is a point other than X₂ =\(1 : 1 : 1\) in the plane of a triangle ABC. Let T denote the triangle with vertices

  \(p : q : r\)

  \(q : r : p\)

  \(r : p : q\)

  Let T' denote the obverse of T, defined in the preamble just before X₂₄₃₀₇ by vertices

  \(p : r : q\)

  \(q : p : r\)

  \(r : q : p\)

  The six points, corresponding to the permutations pqr, qrp, rpq, prq, qpr, rqp, lie on the permutation ellipse of P, as defined in the preamble just before X₃₄₃₄₁, given by

  (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  If P' lies on E(P), then E(P') = E(P). Moreover, if U =\(u : v : w\) is a point, other than P, then the point, other than P', in which the line UP' meets E(P), is the E(P,U)-antipode of P', as defined and formulated in the preamble just before X₃₅₀₂₅.

• X₃₆₂₄₁–X₃₆₂₄₄ — Suren-Moses equilateral-triangle circumcevian-inversion points

  contributed by Peter Moses, January 9, 2020. Suren asked Peter Moses for the locus of a point P such that the circumcevian-inversion triangle of P is equilateral. Moses found that the locus consists of four points, all on the Brocard axis, X₃X₆.

• X₃₆₂₅₆–X₃₆₂₉₅ — TC(X_i,X_j)-antipodes

  contributed by Clark Kimberling and Peter Moses, January 13, 2020. In this paragraph, all coordinates are trilinears. Suppose that P =\(p : q : r\). The trilinear permutation conic denoted by TC(P), is the conic that passes through the six points \(p : q : r\),\(q : r : p\),\(r : p : q\),\(p : r : q\),\(q : p : r\),\(r : q : p\). An equation for TC(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  Thus, TC(P) is analogous, and symbolically identical to, the permutation ellipse E(P), defined in the preamble just before X₃₄₃₄₁.

  Next, suppose that P =\(p : q : r\) (trilinears). For the rest of this paragraph,\(p:q:r\) are trilinears, but all other coordinates, and the equation for TC(P), are in barycentric coordinates. The conic TC(P) passes through the six points \(ap : bq : cr\),\(aq : br : cp\),\(ar : bp : cq\),\(ap : br : cq\),\(aq : bp : cr\),\(ar : bq : cp\). (Note that these six points are not six permutations of \(ap: bq : cr\).) An equation for TC(P) is (q r + r p + p q)(b^2 c^2 x^2 + c^2 a^2 y^2 + a^2 b^2 z^2) - abc(p^2 + q^2 + r^2)(ayz + bzx + cxy) = 0.

  The TC(P,U)-antipode of P is the point, other than P, in which the line PU meets TC(P), where

  P =\(p : q : r\) (trilinears) =\(ap : bq : cr\) (barycentrics)

  U =\(u : v : w\) (trilinears) =\(au : bv : cw\) (barycentrics)

  Barycentrics for TC(P,U)-antipode of P are \(f(a,b,c,p,q,r,u,v,w) : f(b,c,a,q,r,p,v,w,u) : f(c,a,b,r,p,q,w,u,v)\), where (see the preamble).

• X₃₆₃₁₈–X₃₆₄₀₂ — Orthologic centers related to Fermat-Dao-Nhi triangles

  contributed by César Eliud Lozada, January 15, 2020. Fermat-Dao-Nhi equilateral triangles were introduced in the preamble just before X₃₃₆₀₂. These triangles have these properties:

  1st inner-Fermat-Dao-Nhi triangle is the anticomplement of the 3rd outer-Fermat-Dao-Nhi triangle 1st outer-Fermat-Dao-Nhi triangle is the anticomplement of the 3rd inner-Fermat-Dao-Nhi triangle 2nd inner-Fermat-Dao-Nhi triangle is the anticomplement of the 4th outer-Fermat-Dao-Nhi triangle 2nd outer-Fermat-Dao-Nhi triangle is the anticomplement of the 4th inner-Fermat-Dao-Nhi triangle

  A complete list of orthologic and parallelogic centers related to these triangles can be seen here (Attach_X36318.pdf).

• X₃₆₄₀₄–X₃₆₄₁₁ — Centers of TC conics

  contributed by Clark Kimberling and Peter Moses, January 16, 2020. Trilinear permutation conics TC(P) are defined in the preamble just before X₃₆₂₅₆. Briefly, if P =\(p : q : r\) (trilinears), then TC(P), is the conic that passes through the six points \(p : q : r\),\(q : r : p\),\(r : p : q\),\(p : r : q\),\(q : p : r\),\(r : q : p\). An equation for TC(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  For the equivalent formulation using barycentrics, see the aforementioned preamble. If P =\(p : q : r\) (barycentrics), then the center of TC(P) is given by a(2a(q r + r p + p q) + (-a + b + c)(p^2 + q^2 + r^2)) : :

• X₃₆₄₁₂–X₃₆₄₃₂ — Points on the barycentric square of the Euler line

  contributed by Clark Kimberling and Peter Moses, January 16, 2020. Let L denote the Euler line and L^2 the set of barycentric squares of points on L, as in the preamble just before X₂₃₅₈₂. The set L^2 is here named the barycentric Euler inellipse. It has perspector X₂₃₅₈₂ and center X₂₃₅₈₃, and it passes through X_i for these 26 indices i: 2,393,577,3163,7054, and 36412, 36413, 36414, …, 36432.

• X₃₆₄₃₆–X₃₆₄₇₂ — Homothetors involving triangles T(k)

  contributed by Clark Kimberling and Peter Moses, January 17, 2020, and Randy Hutson, January 29, 2020. Suppose that ABC is a triangle. The trisectors of segment BC are \(0:1:2\) and \(0:2:1\); these are two of the points on the permutation ellipse E(\(0:1:2\)), here named the trisection ellipse, given by the equation

  5(x^2 + y^2 + z^2) - 2(y z + z x + x y) = 0.

  For every real number k, let T(k) denote the central triangle with A-vertex \(1 : k : k\). The line AG, where G =\(1:1:1\) = X₂ meets the trisection ellipse in two points,\(1 : k : k\), where k = \(\sqrt{27}\) - 5 and k = - \(\sqrt{27}\) - 5. For these two values of k, the triangle T(k) is homothetic to many triangles, of which 17 for each k give homothetors (centers of homothety) included in this section:

  Euler; (a^2+b^2-c^2) (a^2-b^2+c^2)-2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) k : :

  reflection of ABC in X₃: 2 a^2 (a^2-b^2-c^2)-(a^2+b^2-c^2) (a^2-b^2+c^2) k : :

  reflection of X₃ in ABC; a^2 (a^2-b^2-c^2)+(a^4-3 a^2 b^2+2 b^4-3 a^2 c^2-4 b^2 c^2+2 c^4) k : :

  reflection of ABC in X₅ (aka Carnot, Johnson); a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4-a^2 (a^2-b^2-c^2) k : :

  outer Garcia; b+c+a k : :

  Mandart-incircle triangle; (a-b-c) (a^2-(b-c)^2 k) : :

  inner Yff; 2 a^2 b c+(a^4-2 a^2 b^2+b^4-2 a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4) k : :

  outer Yff; 2 a^2 b c-(a^4-2 a^2 b^2+b^4-2 a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4) k : :

  anti-Aquila; a-(2 a+b+c) k : :

  infinite altitude; (a^2+b^2-c^2) (a^2-b^2+c^2)-2 a^2 (a^2-b^2-c^2) k : :

  3rd tri-squares central; a^2+S-k (a^2+2 S) : :

  4th tri-squares central; a^2-S-k (a^2+2 S) : :

  Ehrmann mid-triangle; a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4-(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) k : :

  anti-inner-Grebe; a^2-k (a^2-S) : :

  anti-outer-Grebe; a^2-k (a^2+S) : :

  1st Kenmotu free-vertices triangle; a^2 (a^2-b^2-c^2+2 S)+(a^4-3 a^2 b^2+2 b^4-3 a^2 c^2-4 b^2 c^2+2 c^4-2 a^2 S) k : :

  2nd Kenmotu free-vertices triangle; a^2 (a^2-b^2-c^2-2 S)+(a^4-3 a^2 b^2+2 b^4-3 a^2 c^2-4 b^2 c^2+2 c^4+2 a^2 S) k : :

  For barycentrics and references for the various triangles, see Index of Triangles Referenced in ETC, by (IndexOfTrianglesReferencedInETC.html).

  For every k, the homothetor of T(k) with each of the following triangles lies on the Euler line: Euler, reflection of X₃ in ABC, reflection of ABC in X₅, infinite altitude.

  For every k, the homothetor of T(k) with each of the following triangles lies on the line X₂X₆: 3rd tri-squares central, 4th tri-squares central, anti-inner Grebe, anti-outer Grebe.

  For every k, the homothetor of T(k) with each of the following triangles lies on the line X₁X₂: outer Garcia, inner Yff, outer Yff, anti-Aquila.

  For k a nonconstant function symmetric in a,b,c, see the preamble just before X₃₆₄₇₃.

  The trisection ellipse is also the conic Cpar(X₂); see the preamble before X₁₀₀₀₁. (Randy Hutson, March 29, 2020)

• X₃₆₄₇₃–X₃₆₅₁₃ — Homothetors involving triangles T(k)

  contributed by Clark Kimberling and Peter Moses, January 18, 2020. In the preamble just before X₃₆₄₃₆, the triangle T(k), for k any real number, is defined as the central triangle with A-vertex \(1 : k : k\). This definition includes triangles for which k is a function symmetric in a,b,c and homogeneous of degree 0, such as (a^2+b^2+c^2)/(bc+ca+ab) and (a^3+b^3+c^3)/(abc).

  For barycentrics and references for the various triangles mentioned in this section, see Index of Triangles Referenced in ETC, (IndexOfTrianglesReferencedInETC.html).

• X₃₆₅₁₄–X₃₆₅₁₇ — V transforms on the circumcircle

  Vu Thanh Tung; related centers X₃₆₄₃₆-X₃₆₄₇₂ were contributed by by Peter Moses, January 20, 2020. Let X =\(x : y : z\) be a point in the plane of a triangle ABC, let A'B'C'= circumcevian triangle of X, and let O_A = circumcenter of triangle XBC; define O_B and O_C cyclically. The triangles O_AO_BO_C and A'B'C' are perspective, and their perspector, on the circumcircle, is given by

  V(X) = a^2 / (a^4 y (y - z) z + (b^2 - c^2) x^2 (c^2 y + b^2 z) + a^2 (-b^2 z (x^2 + 2 x z + y (y + z)) + c^2 y (x^2 + 2 x y + z (y + z)))) : :

  Let X* denote the isogonal conjugate of X; then V(X*) = V(X), as in the following examples:

  V(X₂) = V(X₆) = X₁₂₉₆

  V(X₃) = V(X₄) = X₁₁₀

  V(X₅) = V(X₅₄) = X₁₂₉₁

  V(X₇) = V(X₅₅) = X₂₀₂₁₉

  V(X₉) = V(X₅₇) = X₂₈₂₉₁

  V(X₁₇) = V(X₆₁) = X₃₆₅₁₄

  V(X₁₈) = V(X₆₂) = X₃₆₅₁₅

  V(X₁₉) = V(X₆₃) = X₃₆₅₁₆

  V(X₃₉) = V(X₈₃) = X₃₆₅₁₇

  V(X) is the perspector of ABC and the triangle formed by reflecting line XX' in the sides of ABC, where X' denotes the isogonal conjugate of X. (Randy Hutson, March 29, 2020)

• X₃₆₅₂₁–X₃₆₅₂₅ — Points on the Steiner Midellipse

  contributed by Clark Kimberling and Peter Moses, January 21, 2020. The ellipse midway between the Steiner inellipse (SIE) and the Steiner circumellipse (SCE) is here named the Steiner Midellipse (SME). Specifically, for any X on SCE, let U = segment GX ^ CIE

  . Let M = midpoint of XU

  Then SME is the locus of M as X goes around SCE MSE is, like SIE and SCE, a permutation ellipse; ie., if P = pqr =\(p : q : r\) is on SME, then all six permutations, pqr, qrp, rpq, prq, qpr, rqp are on SME.

  Let G = X₂ = centroid of ABC. If P is on SCE then the point given by the combo G + 3 P is on SME; likewise, if P is on SIE, then G - 3 P is on SIE. An equation for SME follows: 7 (x^2 + y^2 + z^2) - 34 (y z + z x + x y) = 0.

• X₃₆₅₂₆–X₃₆₅₈₇ — Homothetors involving triangles T(k)

  contributed by Clark Kimberling and Peter Moses, January 22, 2020. In the preamble just before X₃₆₄₃₆, the triangle T(k), for k any real number, is defined as the central triangle with A-vertex \(1 : k : k\). This definition includes triangles for which k is a function symmetric in a,b,c and homogeneous of degree 0, such as (bc+ca+ab)/(a^2+b^2+c^2) and abc/(a^3+b^3+c^3). See also the preamble just before X₃₆₄₇₃. For barycentrics and references for the various triangles mentioned in this section, see Index of Triangles Referenced in ETC (IndexOfTrianglesReferencedInETC.html).

• X₃₆₅₉₈–X₃₆₆₅₀ — Cevian-circumconic triangles

  contributed by César Eliud Lozada, January 23, 2020. Let :g:`ABC << A B C >>` be a triangle, P a point, :g:`A’B’C’ << A’ B’ C’ >>` the :term:`cevian triangle <cevian-triangle>` of :g:`P <P>` and K a conic through :g:`A’ <A’>`, :g:`B’ <B’>`, :g:`C’ <C’>`. If :g:`A″ <A″>`, :g:`B″ <B″>`, :g:`C″ <C″>` are the points, others than :g:`A’ <A’>`, :g:`B’ <B’>`, :g:`C’ <C’>`, at which K cuts :g:`BC <[ B C ]>`, :g:`CA <[ C A ]>`, :g:`AB <[ A B ]>`, respectively, then :g:`AA″ <[ A A″ ]>`, :g:`BB″ <[ B B″ ]>`, :g:`CC″ <[ C C″ ]>` are concurrent.

  If P_k is the perspector of K with respect to ABC, the triangle A″B″C″ is named here the (P, P_k)-cevian-circumconic triangle.

  If P =\(x : y : z\) and P_k =\(x_{k} : y_{k} : z_{k}\) (barycentrics) then:

  A″ = 0 : 1/(z*(\(x_{k}\)*y*z+x*y*\(z_{k}\)-3*x*\(y_{k}\)*z)) : 1/(y*(x*\(y_{k}\)*z+\(x_{k}\)*y*z-3*x*y*\(z_{k})\))

  The perspector Q(P, P_k) of ABC and A″B″C″ is:

  Q(P, P_k) = x*(\(x_{k}\)*y*z+x*y*\(z_{k}\)-3*x*\(y_{k}\)*z)*(x*\(y_{k}\)*z+\(x_{k}\)*y*z-3*x*y*\(z_{k})\) : :

  If P_k = P then Q(P, P_k) = P = P_k.

  If P = X₂ then Q(P, P_k) is the isotomic conjugate-of-the anticomplement-of-the anticomplement-of-P_k.

  As a cevian triangle with respect to ABC, A″B″C″ is perspective to these named anticevian triangles: anticomplementary, Bevan antipodal, excentral, Pelletier, Schroeter, Soddy, tangential, X-parabola-tangential.

• X₃₆₆₅₁–X₃₆₆₆₆ — Homothetors involving the Euler triangle and triangles T(k)

  contributed by Clark Kimberling and Peter Moses, January 24, 2020. In this section, k is a quotient of symmetric functions of homogeneity degree 2. The Euler triangle is homothetic to each triangle T(\(1 : k : k\)), with homothetor on the Euler line. See the preambles just before X₃₆₄₃₆ and X₃₆₄₇₃.

• X₃₆₆₇₀–X₃₆₆₉₅ — Homothetors involving the Euler triangle and triangles T(k)

  contributed by Clark Kimberling and Peter Moses, January 25, 2020. In this section, k is a quotient of symmetric functions of homogeneity degree 2. The Euler triangle is homothetic to each triangle T(\(1 : k : k\)), with homothetor on the Euler line. See the preambles just before X₃₆₄₃₆, X₃₆₄₇₃, and X₃₆₆₅₁.

• X₃₆₆₉₇–X₃₆₇₁₆ — Homothetors involving the infinite altitude triangle and triangles T(k)

  contributed by Clark Kimberling and Peter Moses, January 27, 2020. In this section, k is a quotient of symmetric functions of homogeneity degree 2. The infinite altitude triangle is homothetic to each triangle T(\(1 : k : k\)), with homothetor on the Euler line. See the preambles just before X₃₆₄₃₆, X₃₆₄₇₃, and X₃₆₆₅₁.

• X₃₆₇₁₈–X₃₆₇₃₄ — Homothetors involving the Ehrmann mid-triangle and triangles T(k)

  contributed by Clark Kimberling and Peter Moses, January 28, 2020. In this section, k is a quotient of symmetric functions of homogeneity degree 2. The Ehrmann mid-triangle is homothetic to each triangle T(\(1 : k : k\)), with homothetor on the Euler line. See the preambles just before X₃₆₄₃₆, X₃₆₄₇₃, and X₃₆₆₅₁.

• X₃₆₇₆₁–X₃₆₇₈₈ — Largest-circumscribed-equilateral triangle

  contributed by César Eliud Lozada, February 6, 2020. Consider all equilateral triangles A_eB_eC_e circumscribing ABC and such that A lies between B_e and C_e ⁽¹⁾, B lies between C_e and A_e ⁽²⁾ and C lies between A_e and B_e ⁽³⁾ (see note at the end of this preamble). The A-vertex of the largest A_eB_eC_e is the antipode of X₁₃ in the circle {{X₁₃, B, C}}, and the other two vertices are found cyclically. If this triangle is denoted as A'B'C' then A' has barycentric coordinates:

  A' = -6*\(\sqrt{3}\)*S*a^2 - (3*(a^2 + b^2 + c^2))*a^2 + 2*(b^2 - c^2)^2 :

  (7*b^2 + 2*c^2)*a^2 - 3*b^4 + 5*b^2*c^2 - 2*c^4 + 2*\(\sqrt{3}\)*S*(2*a^2 + b^2) :

  (7*c^2 + 2*b^2)*a^2 - 3*c^4 + 5*c^2*b^2 - 2*b^4 + 2*\(\sqrt{3}\)*S*(2*a^2 + c^2)

  The center of A'B'C' is X₅₄₆₃ and its squared-sidelength is 4*S*(cot(ω)+\(\sqrt{3}\))/3, where S and ω are double-area and Brocard angle of ABC, respectively.

  A'B'C' is perspective to the ABC-X₃-reflections-triangle and it is also homothetic to the other triangles in the following list, where the given number n means that the respective homothetor is X(n):

  (ABC-X3 reflections, 36761), (Bankoff, 36762), (3rd Fermat-Dao, 36763), (7th Fermat-Dao, 36764), [and others]

  Orthologic triangles to A'B'C' and orthologic centers:

  (ABC, 5473, 13), (ABC-X3 reflections, 5473, 5473), (anti-Aquila, 5473, 11705), (anti-Ara, 5473, 12142), [and others]

  Parallelogic triangles to A'B'C' and parallelogic centers:

  (2nd Fermat-Dao, 36773, 25216), (4th Fermat-Dao, 6777, 16530), (6th Fermat-Dao, 36773, 25229), [and others]

  Note: The centers of circles {{X₁₃, B, C}}, {{X₁₃, C, A}} and {{X₁₃, A, B}} are the vertices of outer-Napoleon triangle, i.e., A'B'C' is the reflection triangle of X₁₃ in the vertices of the outer-Napoleon triangle. A similar construction can be made using X₁₄ and the inner-Napoleon triangle, but in this case, \(conditions ^{(1)}\), ⁽²⁾, ⁽³⁾ are not all satisfied at the same time.

• X₃₆₇₈₉–X₃₆₇₉₃ — Points on the dual of the circumcircle

  contributed by Clark Kimberling and Peter Moses, February 9, 2020. p> Suppose that P =\(p : q : r\) (barycentrics) is a point on the line at infinity, and U =\(u : v : w\) is a point. Then the point D(P,U) = \(p^{2}\)u : \(q^{2}\)v : \(r^{2}\)w is on the inconic with perspector U. In particular, if U = X₇₆, then D(P,U) lies on the inellipse having perspector X₇₆ and center X₁₄₁. This inellipse is the dual of the circumcircle.

  Also, D(P,X₇₆) is the barycentric quotient P*/P, where P*, the isogonal conjugate of P, lies on the circumcircle.

  The appearance of (i,j) in the following list means that D(X_i,X₇₆) = X_j: (30,36789), (511,36790), (512, 3124), (513,1086), (514,23989), (517,26611), (518,4437), (519,36791), (521,23983), (522,23978), (523,338), (524,36792), (525,36793), (3900,23970)

  The dual of the circumcircle is the barycentric square of line X₅₁₄X₆₆₁ (the trilinear polar of X₇₅). (Randy Hutson, March 29, 2020)

• X₃₆₈₁₀–X₃₆₈₁₃ — Points on the dual of the incircle

  contributed by Clark Kimberling and Peter Moses, February 15, 2020. Suppose that P =\(p : q : r\) (barycentrics) is a point on the line at infinity, and U =\(u : v : w\) is a point. Then the point D(P,U) = q r u : r p v : p q w lies on the circumconic with perspector U. In particular, if U = X₈, then D(P,U) lies on the circumconic having perspector X₈ and center X₃₁₆₁. This circumconic is the dual of the incircle. Also, D(P,X₈) is the barycentric quotient X₈/P.

• X₃₆₈₁₄–X₃₆₈₃₁ — Perspectors associated with mid-trace triangles

  contributed by Clark Kimberling and Peter Moses, February 21, 2020. Let P =\(p : q : r\) and U =\(u : v : w\) be points not on the sidelines BC, CA, AB of a triangle ABC. Let A' = AP∩BC, A' = AU∩BC', and A* = midpoint of A' and A″. Define B* and C* cyclically. The triangle A*B*C* is here named the mid-trace triangle of P and U, denoted by M(P,U).

  A* = 0 : 2 q v + r v + q w : 2 r w + r v + q w

  B* = 2 p u + p w + r \(u : 0 : 2\) r w + p w + r u

  C* = 2 p u + q u + p v : 2 q v + q u + p v

  For given P, the locus of a point X =\(x : y : z\) such that M(P,X) is a cevian triangle is given by the cubic

  p (q r + r^2 + q p) y^2 z - p (q r + q^2 + r p) y z^2 + (cyclic) + (p - q) (p - r) (q - r) x y z = 0,

  here named the mid-cevian cubic of P, denoted by MC(P)

  If P is on the line at infinity, then MC(P) is the union of the line at infiniity (\(x + y + z = 0\)) and the circumconic

  \(p^{2}(q - r)\) y z + \(q^{2}(r - p)\) z x + \(r^{2}(p - q)\) x y = 0

  The following points lie on MC(P): A, B, C, P, 1/p : : , p - 2q - 2r : : 1/r, and p(- p + q + r) : :

  For further developments, see Bernard Gibert’s page, CL069 Mid-Cevian Cubics.

• X₃₆₈₄₉–X₃₆₈₆₉ — Points on mid-cevian cubics

  contributed by Clark Kimberling and Peter Moses, February 21, 2020. The family of mid-cevian cubics is introduced just before X₃₆₈₁₀; specifically, if P is not on BC or CA or AB, then the cubic MC(P) is given by

  p (q r + r^2 + q p) y^2 z - p (q r + q^2 + r p) y z^2 + (cyclic) + (p - q) (p - r) (q - r) x y z = 0.

• X₃₆₈₇₁–X₃₆₉₀₁ — Perspectors associated with Gemini triangles

  contributed by Clark Kimberling and Peter Moses, February 29, 2020. Many triangles, including Gemini triangles 1-111, are itemized in Index of Triangles Referenced in ETC (IndexOfTrianglesReferencedInETC.html). Gemini triangles 112-119 are introduced here by A-vertex, A', as follows:

  Gemini triangle 112: A' = a^2 : b^2 - c^2 : c^2 - b^2

  Gemini triangle 113: A' = a^2 : c^2 - b^2 : b^2 - c^2 (see note below)

  Gemini triangle 114: A' = b \(c : a(b - c) : a(c - b)\)

  Gemini triangle 115: A' = b \(c : a(c - b) : a(b - c)\)

  Gemini triangle 116: A' = a^\(2 : 2(b^{2} - c^{2}) : 2(c^{2} - b^{2})\)

  Gemini triangle 117: A' = a^\(2 : 2(c^{2} - b^{2}) : 2(b^{2} - c^{2})\)

  Gemini triangle 118: A' = 2 a : b - c : c - b

  Gemini triangle 119: A' = 2 a : c - b : b - c

  Note: Gemini triangle 113 is the orthic triangle of the anticomplementary triangle.

  The appearance of (T, i) in the following list means that Gemini triangle 112 is perspective to T, and the perspector is X_i [list omitted here].

  The appearance of (T, i) in the following list means that Gemini triangle 113 is perspective to T, and the perspector is X_i [list omitted here].

• X₃₆₉₂₆–X₃₆₉₃₈ — Points on mid-cevian cubics

  contributed by Clark Kimberling and Peter Moses, March 1, 2020. The mid-cevian cubic MC(P) of a point P is defined in the preamble just before X₃₆₈₁₀.

• X₃₆₉₄₇–X₃₆₉₅₇ — Points of the cubic K1151

  contributed by Peter Moses, March 5-6, 2020. For the definition and properties of K1151, see K1151.

• X₃₆₉₅₈–X₃₇₀₀₈ — Centers and perspectors of Moses Conics

  contributed by Clark Kimberling, March 10-11, 2020. Suppose that U =\(u : v : w\) is a point. The Moses conic of U, denoted by M(U), is here defined by the barycentric equation

  \(vwx^{2}\) + \(wuy^{2}\) + \(uvz^{2}\) - u(v+w)yz - v(w+u)zx - w(u+v)xy = 0

  Suppose that P =\(p : q : r\) lies on M(U). Let CM(P) denote the circumconic of the medial triangle that has center P. Theorem: if P lies on M(U), then U lies on CM(P).

  The center of M(U) is 2u + v + w : 2v + w + u : 2 w + u + v .

  The perspector of M(U) is u/(\(u^{2}\) + vw + 3wu + 3uv) : v/(\(v^{2}\) + wu + 3uv + 3vw) : w/(\(w^{2}\) + uv + 3vw + 3uv).

  The appearance of (n,CM) in the following list means that the point U = X_n lies on the conic CM for every P on M(X_n):

  (1, circumellipse of medial and incentral triangles); see X₃₄₅₈₅)

  (2, Steiner inellipse)

  (3, Johnson circumconic of medial triangle); see K714

  (4, nine-point circle)

  (and others)

• X₃₇₁₂₈–X₃₇₁₄₂ — Always-orthologic triangles

  contributed by César Eliud Lozada, March 11, 2020. Let P =\(x:y:z\) (barycentrics) be a point on the plane of ABC. Then, for any point P, every pair of triangles listed in the following table are orthologic with the given orthologic centers [table omitted here]:

• X₃₇₇₄₃–X₃₇₇₄₆ — Points on the circumellipse with center X₉

  based on notes from Peter Moses, March 14, 2020. Dan Reznik has shown that the circumellipse ellipse with center X₉ has many interesting geometric properties. As stated at X₉, Let E be the circumellipse of T = ABC with center X₉. Then ABC is a billiard orbit of E(3-periodic). If we fix E in the plane, all its triangular orbits (a set of “rotating” triangles T) have the same X₉. Note that X₉ is the point of concurrence of lines drawn from each excenter to the midpoint of the corresponding side of T. (Dan Reznik, June 30, 2019) See [1] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X₉ is stationary at the origin and [2] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X₉ is stationary at the origin. A particularly fine article is recommended: ‘Can the Ellliptic Billiard Still Surprise Us?’, by Dan Reznik, Ronaldo Garcia, and Jair Koiller, in Mathematical Intelligencer 42 (2020) 6-17. An online pdf is available: Click here.

  Certain lines are mapped onto E by barycentric quotients:

  If P lies on the line at infinity, then X₁/P lies on E.

  If P lies on the Euler line, then X₁/(P*X₅₂₅) lies on E.

  If P lies on the Brocard axis, then X₁/(P*X₈₅₀) lies on E.
  If P lies on the circumcircle, then P/X₁ lies on E.

  E = isogonal conjugate of the antiorthic axis.

  E = trilinear pole of the line X₁P.

  For further details regarding the circumellipse E, see X₉. For more points on E, see X₃₇₂₀₂-X₃₇₂₂₃.

• X₃₇₇₅₆–X₃₇₈₀₅ — Cyclologic centers

  Contributed by Emmanuel José Garcia, March 27 - April 1, 2020. Let ABC be a triangle. Consider two points, P and Q, in the plane of ABC. Let P_aP_bP_c and Q_aQ_bQ_c be the pedal triangles of P and Q, respectively. Let X, Y and Z be the orthogonal projections of P onto the sides Q_aQ_b, Q_bQ_c and Q_aQ_c, respectively. Then, P_aP_bP_c and XYZ are cyclologic triangles. For a proof, see GeoDom: Cyclologic triangles.

  The appearance of (i,j,k) in the following list means that X_k is the cyclologic center of the pedal triangle of X_i with respect to X_j: (4,3,10151), (5,3,10615), (3,4,10257), (3,1,37743), (3,4,37744), (3,2,37745), (6,2,37746). (Peter Moses, April 3, 2020)

• X₃₇₈₀₆–X₃₇₈₃₅ — Vu Poles

  based on notes contributed by Vu Thanh Tung, April 5, 2020. Suppose that P =\(p : q : r\) and U =\(u : v : w\) are distinct points in the plane of a triangle ABC. Let

  A₀ = point of intersection, other than P, of the circles (PBC) and (PUA)

  A₁ = AP∩BC

  A₂ = point of intersection, other than A₁, of the circle (PA₀A₁) and line BC, and define B₂ and C₂ cyclically

  The points A₂,B₂,C₂ are collinear; denote their line by v(P,U). Let

  V(P,U) = trilinear pole of v(P,U). The point V(P,U) is here named the Vu pole of P and U.

  Let P* and U* denote the isogonal conjugates of P and U, respectively. Then V(P,U) = V(U*,P*).

  Examples:

  V(X₁,X₃) = X₃₂₁₈

  V(X₁,X₄) = V(X₃,X₁) = X₁₇₉₂₃

  V(X₁,X₅) = X₂₄₁₄₅

  V(X₁,X₆) = V(X₂,X₁) = X₇₂₉₂

  V(X₁,X₉) = X₂₆₀₁₅

  V(X₂,X₄) = V(X₃,X₆) = X₄₆₈

  V(X₂,X₆) = V(X₃,X₆) = X₁₁₅₈₀

  [and others]

  Peter Moses (April 5, 2020) found that

  V(P, U) = q r (a^2 (q r u (u+v+w) - p v w (p+q+r)) - b^2 p u (w (p+q) - r (u+v)) - c^2 p u (v (p+r) - q (u+w))) : :

  and that if U is on the Euler line, then V(X₂,U) is also on the Euler line. Specifically, if U is given by the combo X₂ + k X₃, then

  V(X₂, U) = (-a^2 - b^2 + c^2)(a^2 - b^2 + c^2)(-a^2 + b^2 + c^2) + a^2 (3 a^4 - 3 b^4 + 4 b^2 c^2 - 3 c^4) + 3 k a^2 (a^4 - b^4 + b^2 c^2 - c^4) : :

• X₃₇₈₄₁–X₃₇₈₄₅ — Centers of Vu circles

  based on notes contributed by Vu Thanh Tung, April 6, 2020. Suppose that P =\(p : q : r\) and U =\(u : v : w\) are distinct points in the plane of a triangle ABC. Let

  A₀ = AU∩BC

  A₁ = point of intersection, other than P, of the line PA and the circle (PBC)

  A₂ = point of intersection, other than A₁, of the line A₀A₁ and the circle (PBC)

  The points P, A₂,B₂,C₂ are concyclic; denote their circle by c(P,U). This circle is here named the Vu circle of P and U; see Theorem 1 (Theorem1Vu.pdf). The center of c(P,U), denoted by VT(P,U), is given by [long equation, omitted here].

  Special cases include

  VT(X₁,U) = a (a^2 v w - b c u (v + w) - a u (c v + b w)) : :

  VT(X₄,U) = -a^6 v w + a^4 (b^2 + c^2) v w + (b^2 - c^2)^2 u (c^2 v + b^2 w) - a^2 (b^2 + c^2) u (c^2 v + b^2 w) : :

  Examples:

  VT(X₁,X₂) = X₁₀₀₁

  VT(X₁,X₃) = X₁₀₅₇₁

  VT(X₁,X₄) = X₁₆₅₀₂

  VT(X₁,X₆) = X₉₉₅

  VT(X₁,X₇) = X₉₉₉

  VT(X₁,X₈) = X₃

  VT(X₁,X₉) = X₃₅₇₆

  VT(X₁,X₁₀) = X₂₁

  VT(X₃,X₆) = X₆₆₄₄

  VT(X₄,X₁) = X₃₅₅

  VT(X₄,X₂) =X₁₃₅₂

  VT(X₄,X₃) = X₃

  VT(X₄,X₅) =X₆₂₈₈

  VT(X₄,X₆) =X₃₈₁

  The appearance of (i,j,k) in the following list means that VT(X_i,X_j) = X_k: (1,5,37806), (2,1,37807), (2,3,37808), (2,4,37809), (2,5,37810), (2,6,37811), (3,1,37812), (3,2,37813), (3,4,37814), (1,11,37815), [and others]

• X₃₇₈₆₉–X₃₇₈₈₀ — Antigonal images

  based on notes contributed by Vu Thanh Tung, April 9, 2020. Let P =\(p:q:r\) (barycentrics) be a point in the plane of a triangle ABC. Let A' be the point, other than P, in which the line AP meets the circle (PBC), and define B' and C' cyclically; the triangle A'B'C' is called the circlecevian triangle of P with respect to triangle ABC by Floor van Lamoen (Hyacinthos #10039); see the preambles just before X₃₄₅₂₀ and X₃₄₈₉₂.

  Assume that P is not X₄, and let A″ = reflection of A' in line BC, and define B″ and C″ cyclically. The lines AA″, BB″,CC″ concur in a point IA(P), and A″B″C″ is the circlecevian triangle of IA(P) with respect to ABC. The point IA(P) is the antigonal image of P; for a definition see the Glossary of ETC, where it is noted that.

  IA(P) = p / ( a^2 (p + q) (p + r) - b^2 p (p + q) - c^2 p (p + r) ) : :

  Note that IA(IA(P)) = P, and that IA(P) lies on the rectangular hyperbola passing through these five points: A, B, C, P, X₄.

  Examples:

  IA(X₁) = X₈₀

  IA(X₂) =X₆₇₁

  IA(X₃) = X₂₆₅

  IA(X₄) (undefined)

  IA(X₅) = X₁₂₆₃

  [and others]

• X₃₇₈₉₆–X₃₇₉₈₀ — Parallels-tangential-conics and related centers

  contributed by César Eliud Lozada, April 15, 2020. Parallels-conics are defined in the preamble just before X₁₀₀₀₁:

  Let A' be the line through a point P parallel to line BC. Let A_B = A'∩AB and A_C = A'∩AC. Define B_C and C_A cyclically, and define B_A and C_B cyclically. The six points A_B, B_C, C_A, A_B, B_C, C_A lie on a conic, here named the parallels-conic of P, denoted by Cpar(P).

  Also, the lines AB_A, AC_A, BC_B, BA_B, CA_C, CB_C are tangent to another conic, here named the parallels-tangential-conic of P and denoted by TCpar(P). If P =\(x : y : z\) (barycentrics), then TCpar(P) has center O(P) and perspector W(P) given by

  O(P) = 2*x^3 + x*y*(5*x + 3*y + 2*z) + x*z*(5*x + 2*y + 3*z) + 2*y*z*(x + y + z) : :

  W(P) = (x + y)*(x + z)*(y*(x + y + z) + 2*x*z)*(z*(x + y + z) + 2*x*y) : :

• X₃₇₉₉₄–X₃₇₉₉₇ — Circumcircle-inverses of points on the Euler line

  contributed by Clark Kimberling and Peter Moses, April 18-19, 2020. If X is a point on the Euler line, then the circumcircle-inverse of X is on the Euler line. In addition to the 4877 triangle centers on the Euler line that are listed in the Central Lines page (accessible via Tables at the top of this page), this section introduces 85 more.

  The appearance of {i,j} in the following list means that X_i and X_j are a pair of circumcircle-inverses on the Euler line: {2,23), (3,30), (4,186), (5,2070), (20,2071), (21,1325), (22,858), (24,403), (25,468), (26,2072), (27,2073), (28,2074), (29,2075), (140,5899), (199,33329), (235,37917), (237,1316), (297,36176), (376,7464), (378,10295), (381,7575), (382,15646), (384,37896), (401,37918), (404,37919), (427,21284), (428,37920), [and others].

Part 20

• X₃₈₀₀₁–X₃₈₀₀₂ — Stationary points of Hutson right hyperbolas

  conributed by Clark Kimberling, based on notes from Randy Hutson and Peter Moses, April 26, 2020. The following note is copied from X₃₇₉₀₄. “Let P and P' be circumcircle antipodes. Let Q be the midpoint of X₂ and P. Let Q' be the midpoint of X₂ and P'. The rectangular hyperbola passing through P, P', Q, Q' has center X₃₇₉₀₄ for all P. (Randy Hutson, April 24, 2020)”.

  There are two stationary points on the family of Hutson rectangular hyperbolas, and on the Euler line, as can be seen by dragging P around the circumcircle in this GeoGebra diagram: Hutson Right Hyperbolas (HutsonRightHyperbolas.ggb). (Peter Moses, April 26, 2020)

  Let HRH(P) denote the Hutson right hyperbola of P.

  HRH(X₇₄ passes through X_i for these i: 74, 110, 2574, 2575, 2930, 5642, 5646, 38001, 38002.

  HRH(X₉₈) passes through X_i for these i: 98, 99, 2482, 6055, 38001, 38002.

  HRH(X₁₁₁) passes through X_i for these i: 111, 1296, 9172, 33900, 38001, 38002.

  For P =\(p : q : r\) (barycentrics) on the circumcircle, let

  \(h(a,b,c,p,q,r,x,y,z) = (a^{2} - 5 b^{2} - 5 c^{2}) \cdot (2 a^{2} b^{2} c^{2} p (-q + r) - c^{2} (a^{2} + b^{2} - c^{2}) \cdot (a^{2} + c^{2}) q^{2} - 2 b^{2} (b^{2} - c^{2}) c^{2} q r + b^{2} (a^{2} + b^{2}) \cdot (a^{2} - b^{2} + c^{2}) r^{2}) x^{2} - (a^{4} - b^{4} + 10 b^{2} c^{2} - c^{4}) \cdot ((b^{4} - c^{4}) p^{2} + 2 a^{2} p (b^{2} q - c^{2} r) + a^{2} (a^{2} + c^{2}) q^{2} - a^{2} (a^{2} + b^{2}) r^{2}) y z\)

  An equation for HRH(P) is

  h(a,b,c,p,q,r,x,y,z) + H(b,c,a,q,r,p,y,z,x) + \(H(c,a,b,r,p,q,z,x,y) = 0\).

  The family of Hutson right hyperbolas can be generalized by replacing X₂ in the definition of HRH(P) by U =\(u : v : w\). Let

  \(H(a,b,c,u,v,w,p,q,r,x,y,z) = (c^{2} (a^{2} + b^{2} - c^{2}) \cdot (a^{2} + c^{2}) q^{2} + 2 a^{2} b^{2} c^{2} p (q - r) + 2 b^{2} c^{2} (b^{2} - c^{2}) q r - b^{2} (a^{2} + b^{2}) \cdot (a^{2} - b^{2} + c^{2}) r^{2}) \cdot (c^{2} u v + 2 c^{2} v^{2} + b^{2} u w - (a^{2} - 2 b^{2} - 2 c^{2}) v w + 2 b^{2} w^{2}) x^{2} - ((b^{4} - c^{4}) p^{2} + a^{2} (a^{2} + c^{2}) q^{2} - a^{2} (a^{2} + b^{2}) r^{2} + 2 a^{2} p (b^{2} q - c^{2} r)) \cdot (4 b^{2} c^{2} u^{2} + c^{2} (a^{2} + 3 b^{2} - c^{2}) u v + b^{2} (a^{2} - b^{2} + 3 c^{2}) u w + a^{2} (a^{2} - b^{2} - c^{2}) v w) y z\)

  Then the general right hyperbola is given by

  H(a,b,c,u,v,w,p,q,r,x,y,z) + h(b,c,a,v,w,u,q,r,p,y,z,x) + \(h(c,a,b,w,u,v,r,p,q,z,x,y) = 0\),

  with the following point, independent of P, as center:

  4*a^2*b^2*c^2*u^3 + c^2*(4*a^4 + 3*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4)*u^2*v + a^2*c^2*(3*a^2 + b^2 - 3*c^2)*u*v^2 + b^2*(4*a^4 - 5*a^2*b^2 + b^4 + 3*a^2*c^2 - 2*b^2*c^2 + c^4)*u^2*w + 2*a^2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 - c^4)*u*v*w + 3*a^4*(a^2 - b^2 - c^2)*v^2*w + a^2*b^2*(3*a^2 - 3*b^2 + c^2)*u*w^2 + 3*a^4*(a^2 - b^2 - c^2)*v*w^2 : : ,

  and two stationary points on the line UX₃. The midpoint of the two points is the center of the hyperbola. As an example, for U = X₄, the center is X₄₆₈ and the stationary points are X₅₀₀₀ and X₅₀₀₁, and if P = X₇₄, then the hyperbola is the Walsmith rectangular hyperbola. (Peter Moses, April 26, 2020)

  The general hyperbola is denoted by (U,P)-MHRH and here named the (U,P)-Moses-Hutson right hyperbola. For further examples, see X₃₈₀₁₀-X₃₈₀₁₄.

• X₃₈₀₀₅–X₃₈₀₀₉ — Vu pedal translations

  based on notes from Vu Thanh Tung, April 25, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC, but not on the circumcircle, and not the incenter or any of the three excenters. Let

  P' = isogonal conjugate of P

  A'B'C' = pedal triangle of ABC

  V = vector PP'

  A₁B₁C₁ = V(A'B'C')

  Then A₁B₁C₁ is perspective to ABC, and the perspector, here named the Vu pedal translation of P, is the point

  V(P) = (a^4 q r + (b^2 - c^2) p (c^2 q + b^2 r) + a^2 (c^2 q (p - r) + b^2 r (3 p + 3 q + 2 r)))*(a^4 q r - (b^2 - c^2) p (c^2 q + b^2 r) + a^2 (b^2 (p - q) r + c^2 q (3 p + 2 q + 3 r))) : :

  See Vu Pedal Translation (VuPedalTranslation.pdf).

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (2,38005), (3,4), (4,4846), (5,38006), (6,18842), (7,38007), (8,38008), (9,38009)

  See the preamble just before X₃₈₃₀₅ for Vu antipedal translation.

• X₃₈₀₁₇–X₃₈₀₂₀ — Foci of circumparabolas

  :based on notes from Peter Moses and Randy Hutson, April 28-29, 2020. Suppose that one focus of a circumparabola is P =\(p : q : r\) on the line at infinity. Then the other focus lies on the circumparabolas foci quintic, Q077 and is given by

  p*((-a^2 + b^2 + c^2)*p^2*q*r - a^2*q^2*r^2 + b^2*r^2*p^2 + c^2*p^2*q^2): :

  The circumparabola, given by

  p(q + r) y z + q( r + p ) z x + r( p + q) = 0

  is the isogonal conjugate of the line tangent to the circumcircle at the isogonal conjugate of P.

  The appearance of (i,j) in the following list means that X_i is on the line at infinity and X_j is the corresponding focus of a circumparabola: (30,38246), (512,38017), (513,38018), (514,38019), (523,12064), (524,38020), (525,38233). (Peter Moses, April 29, 2020)

• X₃₈₀₂₁–X₃₈₂₃₂ — Centroids of triangles with central vertices

  contributed by César Eliud Lozada, April 28, 2020. This section includes centroids of triangles {X_i, X_j, X_k}, with i < j< k ≤ 12 and some triangles with vertices X₁₃ to X₁₈. The following list shows centroids of triangles in the above range but not given in this section. The appearance of (i, j, k, n) in this list means that the centroid of trinagle {X_i, X_j, X_k} is X_n: (1, 2, 3, 3653), (1, 2, 8, 19875), (1, 2, 10, 19883), (1, 3, 4, 5886), (1, 3, 8, 26446), (1, 3, 10, 10165), (1, 4, 8, 5587), (1, 4, 10, 3817), (1, 4, 20, 3576), (1, 5, 10, 11230), (1, 8, 10, 10), (1, 8, 11, 34122), (1, 8, 20, 165), (1, 10, 11, 32557), (2, 3, 4, 5055), (2, 3, 5, 11539), [and others]

• X₃₈₂₄₇–X₃₈₃₀₄ — Vu-Lozada QA-points

  contributed by César Eliud Lozada, May 1, 2020. The following theorem is enunciated in the preamble just before X₃₆₅₉₈:

  Let :g:`ABC << A B C >>` be a triangle, P a point, :g:`A’B’C’ << A’ B’ C’ >>` the :term:`cevian triangle <cevian-triangle>` of :g:`P <P>` and ℭ a conic through :g:`A’ <A’>`, :g:`B’ <B’>`, :g:`C’ <C’>`. If :g:`A″ <A″>`, :g:`B″ <B″>`, :g:`C″ <C″>` are the points, others than :g:`A’ <A’>`, :g:`B’ <B’>`, :g:`C’ <C’>`, at which ℭ cuts :g:`BC <[ B C ]>`, :g:`CA <[ C A ]>`, :g:`AB <[ A B ]>`, respectively, then :g:`AA″ <[ A A″ ]>`, :g:`BB″ <[ B B″ ]>`, :g:`CC″ <[ C C″ ]>` concur.

  It is also written in that preamble that if P =\(x : y : z\) and ℭ has perspector X' =\(x' : y' : z'\) with respect to ABC (barycentrics), then the triangles ABC and A″B″C″ have perspector Q(P, X') = x (x’ y z + x (z’ y - 3 y’ z)) (x’ y z + x (y’ z - 3 z’ y)) : :

  Vu Thanh Tung observed that if the previous construction is applied to a quadrangle P₁P₂P₃P₄ and every Q_i is calculated as above in the triangle P_jP_kP_ℓ, then the four lines P_nQ_n are also concurrent (Quadri-and-Poly-Geometry #243). César Lozada found that this point of concurrence T(P, X') has barycentrics coordinates:

  T(P, X') = x’ (x’ y z + x (y’ z - 3 z’ y)) (x’ y z + x (z’ y - 3 y’ z)) : : . The point T(P, X') is nere named the Vu-Lozada QA-point of (P, X’). Notes:

  If P = X' or P lies on the ABC-circumconic with perspector X' then T(P, X') = P. If X' = X₂ then T(P, X') = IsotomicConjugate(Anticomplement(Anticomplement(IsotomicConjugate(P)))). If X' = X₄ then T(P, X') = PolarConjugate(Anticomplement(Anticomplement(PolarConjugate(P)))). If X' = X₆ then T(P, X') = IsogonalConjugate(Anticomplement(Anticomplement(IsogonalConjugate(P)))).

  The appearance of (i, j, n) in the following list means that the Vu-Lozada QA-point of (X_i, X_j) is X_n:

  (1, 2, 38247), (1, 3, 38248), (1, 4, 38249), (1, 6, 3445), (1, 7, 38250), (1, 8, 38251), (1, 31, 38252), (2, 1, 8056), (2, 3, 1073), (2, 4, 38253), (2, 6, 8770), (2, 7, 38254), (2, 8, 38255), (2, 111, 38280), (3, 1, 36600) [and others]

• X₃₈₃₀₅–X₃₈₃₀₉ — Vu antipedal translations

  based on notes from Vu Thanh Tung, May 1, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC, but not on the circumcircle. Let

  A'B'C' = antipedal triangle of P

  V = 2 * vector OP'

  A₁B₁C₁ = V(A'B'C')

  Then A₁B₁C₁ is perspective to ABC, and the perspector, here named the Vu antipedal translation of P, is the point

  V(P) = 1 / ( c^4 q (2 p - r) + b^4 (2 p - q) r - 3 a^4 q r - 2 a^2 (c^2 q (p - 2 r) + b^2 (p - 2 q) r) + 2 b^2 c^2 (q r + p (q + r)) ) : :

  See Vu Antipedal Translation (VuAntipedalTranslation.pdf).

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (1,3577), (2,14484), (4,4), (5,38305), (6,3531), (7,38306), (8,38307), (9,38308), (10,38309)

  See the preamble just before X₃₈₀₀₅ for Vu pedal translation.

• X₃₈₃₁₀–X₃₈₃₁₃ — Points Associated with generalized Paasche conics

  contributed by Clark Kimberling and Peter Moses, May 4, 2020. As stated at X₃₇₈₆₁, if P =\(p : q : r\) and U =\(u : v : w\) (barycentrics) are triangle centers having the same degree of homogeneity in a,b,c, then the (p,u)-generalized Paasche conic, GPC(p,u), is the conic that passes through these six \(points: 0 : r\) : w,\(0 : v : q\),\(u : 0 : p\),\(r : 0 : w\),\(q : v : 0\),\(u : p : 0\) and is given by the equation p v w x^2 + q w u y^2 + r u v z^2 - (q r u + u v w) y z - (r p v + u v w) z x - (p q w + u v w) x y = 0.

  The perspector of GPC(p,u) is the barycentric quotient P/U = p v w : q w u : r u v, and the center of GPC(p,u) is 2 p^2 v^2 w^2 - u v w (2 p q r + 2 q r u + 2 p r v + 2 p q w + r u v + q u w) : :

  Note, for example, that GPC(p,2u) is not the same conic as GPC(p,u); i.e., the definition of generalized Paasche conic depends on the representations of centers P and U. Nevertheless, we write GPC(P,U) instead of GPC(p,u) in cases where p and u are the first barycentrics as shown in ETC; in particular, this is the case when p and u are polynomials in a,b,c with relatively prime coefficients that agree in parity. Examples of this kind include the following:

  Degree 2 of homogeneity:

  GPC(X₉,X₁₉₂) passes through X₁₀₀₃₀.

  GPC(X₇₅,X₉) passes through X₃₃₀₇ and X₃₃₀₈, and has center X₆₅₀.

  GPC(141),X₆) passes through X₁₆₆₂ and X₁₆₆₃, a hyperbola with center X₁₈₂.

  GPC(X₁₉₂,X₉) passes through X₃₆₈₅.

  GPC(X₂₃₉,X₃₂₀) passes through X₃₂₀.

  GPC(X₃₂₀,X₂₃₉) passes through X₂₃₉ and its antipode, X₃₈₃₁₁.

  Degree 3 of homogeneity:

  GPC(X₁₁,X₅₅) passes through X₁₀₁ and X₃₉₃₉, and has center X₁₅₂₆₀.

  GPC(X₁₁,X₁₀₀) passes through X₇₆₅, X₄₅₆₄, and has center X₃₈₃₁₀.

  GPC(X₃₈, X₃₁) = GPC(X₁₄₁,X₆).

  GPC(X₄₂,X₁₇₁) passes through X₄₁₂₈.

  GPC(X₅₅,X₁₁)) passes through X₅₁₄ and X₅₂₂, a hyperbola with center X₁₅₂₈₀

  GPC(X₅₅,X₄₃) passes through X₁₀₀₃₀.

  GPC(X₅₅,X₃₁₂) passes through X₁₉₂₁.

  GPC(X₁₀₀,X₁₁)) passes through X₅₁₄ and X₅₂₂, a hyperbola with center X₁₁

  GPC(X₁₀₀,X₂₄₄) passes through X₅₁₄.

  GPC(X₁₇₁,X₄₂) passes through X₂₆₄₃.

  GPC(X₂₄₄,X₁₀₀) passes through X₁₀₁₆.

  GPC(X₃₂₁,X₂₁₀) = GPC(X₇₅,X₉).

  GPC(X₃₅₄,X₂₁₀) passes through X₃₉₃₂.

  GPC(p,u} passes through \(u : v : w\) if q r + r p + p q + v w + w u + u v = p u + q v + r w.

  GPC(p,u} passes through \(p : q : r\) if q^2 r^2 u + p^2 r^2 v + p^2 q^2 w + (q r + r p + p q) u v w = p^3 v w + q^3 w u + r^3 u v.

  As noted at X₃₇₈₆₁, César Lozada observed that the Paasche inner conic (i.e., the Paasche ellipse), passes through the points indicated by the notation GCP(1,sin A). To generalize, the locus of the center of GCP(t, sinA) as t goes through the real numbers is the quartic curve given by

  a*b*(b - c)^2*c*x^4 + 2*(a - b)*b*(a - 2*c)*(b - c)*c*x^3*y + a*b*c*(a^2 - 4*a*b + b^2 + 2*a*c + 2*b*c - 2*c^2)*x^2*y^2 - 2*a*(a - b)*(b - 2*c)*(a - c)*c*x*y^3 + a*b*(a - c)^2*c*y^4 - 2*(a - 2*b)*b*(a - c)*(b - c)*c*x^3*z - 2*a*b*c*(a^2 - a*b + b^2 - a*c - b*c + c^2)*x^2*y*z - 2*a*b*c*(a^2 - a*b + b^2 - a*c - b*c + c^2)*x*y^2*z + 2*a*(2*a - b)*(a - c)*(b - c)*c*y^3*z + a*b*c*(a^2 + 2*a*b - 2*b^2 - 4*a*c + 2*b*c + c^2)*x^2*z^2 - 2*a*b*c*(a^2 - a*b + b^2 - a*c - b*c + c^2)*x*y*z^2 - a*b*c*(2*a^2 - 2*a*b - b^2 - 2*a*c + 4*b*c - c^2)*y^2*z^2 + 2*a*(a - b)*b*(a - c)*(2*b - c)*x*z^3 - 2*a*(a - b)*b*(2*a - c)*(b - c)*y*z^3 + a*(a - b)^2*b*c*z^4 = 0.

  This curve passes through the points X_i for i = 2, 9, 3218, 8183, 37861, 37862, 38312, 38313.

  The following table identifies the conics GPC(t, sin A) for several choices of t:

  t GPC(t, sin A) center 0 Steiner circumellipse X₂ 1 Paasche elllipse X₃₇₈₆₁ -1 Paasche outer conic X₃₇₈₆₂ infinity circumellipse centered at X₉ X₉ -2/(3R) hyperbola X₃₂₁₈ -W^(1/2) (see below) (pending) X₃₈₃₁₂ W^(1/2) (see below) (pending) X₃₈₃₁₃

  In the table, W = 3S/(4Rs) = 3(-a+b+c)(a-b+c)(a+b-c)/(4 a b c).

• X₃₈₃₁₇–X₃₈₃₁₉ — Centers of Vu pedal-centroidal circles

  based on notes from Vu Thanh Tung, May 4, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. Let

  G = X₂ = centroid of ABC

  A₀B₀C₀ = medial triangle of ABC

  A₁B₁C₁ = pedal triangle of P

  A₂ = centroid of A₁B₀C₀

  B₂ = centroid of B₁A₀C₀

  C₂ = centroid of C₁A₀B₀

  The points G, A₂, B₂, C₂ lie on a circle, here named the Vu pedal-centroidal circle of P. The center of this circle is the point

  V(P) = a^4 (2 p + q + r) + (b^2 - c^2)^2 (3 p + 2 (q + r)) - a^2 (b^2 + c^2) (5 p + 3 (q + r)) : :

  See PedalCentroidCircle (PedalCentroidCircle.pdf)

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (1,11230), (2,15699), (3,2), (4,5), (5,547), (6,38317), (7,38171), (8,38042), (9,38318), (10,10172), (11,38319), (20,549), (74,34128), (98,34127), (140, 3628), (376, 11539), (381, 5055), (382, 381)

  If P* is the circumcircle-inverse of P, then the Vu pedal-centroidal circles of P and P* are tangent at X₂. (Randy Hutson, May 5, 2020)

  If P lies on the Euler line, then V(P) also lies on the Euler line. If P lies on the line at infinity, then V(P) = P. (Randy Hutson, May 5, 2020)

  V maps the circumcircle onto the circle of the points V(X₁₀₄) = X₃₄₁₂₆, V(X₉₈) = X₃₄₁₂₇, and V(X₇₄) = X₃₂₁₂₈; the center of this circle is X₂. (Randy Hutson, May 5, 2020)

• X₃₈₃₃₁–X₃₈₃₃₄ — Vu circumcevian-orthocenter perspectors

  based on notes from Vu Thanh Tung, May 9, 2020. In the plane of a triangle ABC, let P =\(p : q : r\) (barycentrics), and let

  A₁B₁C₁ = circumcevian triangle of P

  H_bc = orthocenter of PB₁C, and define H_ca and H_ab cyclically

  H_cb = orthocenter of PC₁B, and define H_ac and H_ba cyclically

  A' = H_caH_ac∩H_abH_ba, and define B' and C' cyclically.

  Then A'B'C' is perspective to ABC, and the perspector is given by

  V(P) = (2 b^2 c^2 p^2 + a^2 c^2 p q + b^2 c^2 p q - c^4 p q + a^2 b^2 p r - b^4 p r + b^2 c^2 p r + a^4 q r - a^2 b^2 q r - a^2 c^2 q r) * (-a^2 b^2 c^2 p q + b^4 c^2 p q - a^2 c^4 p q - 2 b^2 c^4 p q + c^6 p q - a^4 c^2 q^2 + a^2 b^2 c^2 q^2 - a^2 c^4 q^2 + a^4 b^2 p r - 2 a^2 b^4 p r + b^6 p r - 2 b^4 c^2 p r + b^2 c^4 p r + a^6 q r - 2 a^4 b^2 q r + a^2 b^4 q r - a^4 c^2 q r - a^2 b^2 c^2 q r) * (a^4 c^2 p q + b^4 c^2 p q - 2 a^2 c^4 p q - 2 b^2 c^4 p q + c^6 p q - a^2 b^4 p r + b^6 p r - a^2 b^2 c^2 p r - 2 b^4 c^2 p r + b^2 c^4 p r + a^6 q r - a^4 b^2 q r - 2 a^4 c^2 q r - a^2 b^2 c^2 q r + a^2 c^4 q r - a^4 b^2 r^2 - a^2 b^4 r^2 + a^2 b^2 c^2 r^2) : :

  See Vu Circumcevian-Orthocenter Perspector (VuCircumcevianOrthocenterPerspector.pdf).

  The appearance of (i,j) in the following list means that V(X_i) = X_j: (1,79), (2,38331), (3,3), (4,14111), (5,38332), (6,22100) (7,38333), (8,38334)

• X₃₈₃₃₆–X₃₈₃₄₃ — 2nd Vu circumcevian-orthocenter perspectors and trilinear poles

  based on notes from Vu Thanh Tung, May 10, 2020. As in the preamble just before X₃₈₃₃₁, let P =\(p : q : r\) (barycentrics), and let

  A₁B₁C₁ = circumcevian triangle of P

  H_bc = orthocenter of PB₁C, and define H_ca and H_ab cyclically

  H_cb = orthocenter of PC₁B, and define H_ac and H_ba cyclically

  A' = H_caH_ac∩H_abH_ba, and define B' and C' cyclically.

  Then A'B'C' is perspective to A₁B₁C₁, and the perspector, here named the 2nd Vu circumcevian-orthocenter perspector of P, denoted by V₂2(P). The points P, V(P), V₂(P) are collinear in a line d(P), here named the Vu circumcevian-orthocenter perspectrix. The trilinear pole of d(P) is denoted by T(P). Barycentrics for V(P) are given in the preamble just before X₃₈₃₃₁, and barycentrics for V₂(P) and T(P) are given here (VuBarycentricsPreambleX(38336).txt). See Second Vu Circumcevian-Orthocenter Perspector (SecondVuCircumcevianOrthocenterPerspector.pdf).

  The appearance of (i,j) in the following list means that V₂(X_i) = X_j: (1,38336), (2,38337), (3,3), (4,6242), (6,38339), (25,38338).

  The appearance of (i,j) in the following list means that T(X_i) = X_j: (1,38340), (2,38341), (4,38342), (6,38343)

  If T_X=pedal-of-X and T_U=pedal-of-U, then E(T_X,T_U) is the orthopole of line XU. (Randy Hutson, May 19, 2020)

• X₃₈₃₄₄–X₃₈₄₂₉ — Equicenters: X₃₈₃₄₄-X₃₈₄₂₉

  contributed by César Eliud Lozada, May 10, 2020. Let A'B'C' and A″B″C″ be triangles inscribed in ABC. The affine transformation sending A' to A″, B' to B″, C' to C″ has a fixed point E named the equicenter of triangles A'B'C' and A″B″C″. (Reference: The Triangle Web by Quim Castellsaguer).

  The fixed point E is unique, and if T' and T″ are homothetic, then E is their homothetic center.

  The equicenter of the affine transformation sending (A', B', C') to (A″, B″, C″) coincides with the equicenter of the inverse affine transformation sending (A″, B″, C″) to (A', B', C'). Therefore, the equicenter of T' and T″ may be referred simply as the equicenter of triangles T’ and T”, regardless of the order in which the triangles are listed.

  The equicenter of T' and T″ is also the similarity image of T' and T″..

  Some particular results:

  If normalized barycentric coordinates of A', B', C' are A' =\(x_{a} : y_{a} : z_{a}\) and B' =\(x_{b} : y_{b} : z_{b}\) : C' =\(x_{c} : y_{c} : z_{c}\), then the equicenter of ABC and A'B'C' is given by

  Ed(ABC, A'B'C') = \(x_{b}\) \(x_{c}\) + \(x_{b}\) \(y_{c}\) + \(x_{c}\) \(z_{b}\) : :

  Suppose that X =\(x : y : z\) and U =\(u : v : w\) (barycentrics). If T_X = cevian-of-X and T_U = cevian-of-U, then E(T_X, T_U) = x u (y w - z v) ((y - z) u - (x + z) v + (x + y) w) : :

  For a list of equicenters related to ABC see here (./X38344List.txt). Also, definitions of triangles mentioned can be found in the index of triangles (IndexOfTrianglesReferencedInETC.html).

  Open problem: give a geometric construction of the equicenter of two arbitrary triangles.

  Indeed, on October 29, 2005, François Rideau provided and proved a method, based strictly on intersections of lines, for constructing the fixed point of an affine transformation (see Francois Rideau - Les points fixes d’une application affine.pdf (Francois%20Rideau%20-%20Les%20points%20fixes%20d%27une%20application%20affine.pdf) (in French)). This is his construction:

  Given two non-homothetic triangles :g:`A’B’C’ << A’ B’ C’ >>` and :g:`A″B″C″ << A″ B″ C″ >>` and the affine transformation ƒ({ A', B', C' }) → { A″, B″, C″ }, we complete the parallelograms A’B’:g:`C’D’ <[ C’ D’ ]>` and A”B”:g:`C″D″ <[ C″ D″ ]>`; i.e., :g:`D’ <D’>` is the reflection of :g:`B’ <B’>` in the midpoint of segment :g:`A’C’ <[ A’ C’ ]>` and :g:`D″ <D″>` is the reflection of :g:`B″ <B″>` in the midpoint of segment :g:`A″C″ <[ A″ C″ ]>`. Let A* = A'B' ∩ A″B″, B* = B'C' ∩ B″C″, C* = C'D' ∩ C″D″, D* = D'A' ∩ D″A″. Then the fixed point M of the affine transformation ƒ is M = A*C* ∩ B*D*.

  Additionally, Rideau provided a very simple method for finding ƒ(X) of a given point X: let U = A*B* ∩ parallelLine(X, A'B') and V = B*D* ∩ parallelLine(X, A'D'). Then ƒ(X) = parallelLine(U, A″B″) ∩ parallelLine(V, A″D″).

  Many thanks to Francisco Javier García Capitán for his notes with simplifications of Rideua conclusions and to Angel Montesdeoca for sending me these notes. (César Lozada, January 26, 2021.)

  If T_X=pedal-of-X and T_U=pedal-of-U, then E(T_X,T_U) is the orthopole of line XU. (Randy Hutson, May 19, 2020)

• X₃₈₄₃₃–X₃₈₄₄₉ — Points associated with Vu (k)-conics

  based on notes from Vu Thanh Tung, May 13-14, 2020 and Peter Moses, May 13, 2020. Suppose that 0 < k < π/2. Let A' and A″ be the points on line BC such that |AA'| = |AA″| and angle A'AA″ has measure 2k. Likewise, let B' and B″ on CA and C' and C″ on AB be the points such that the triangles B'BB″ and C'CC″ are similar to A'AA″. Then the points A', A″, B', B″, C', C″ lie on a conic, here named the Vu (k)-conic. Peter Moses found that these conics all have center X₆ and that the perspector, V(k), of the Vu (k)-conic is given by

  V(k) = 1/((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Cos[k]^2 - 4*(a^4 - b^4 - c^4)*S^2*Sin[k]^2) : : (barycentrics)

  Moses also found that he ratio of minor axis to major axis is \(\sqrt{(J-1) (J+3) / ((J-3) (J+1))}\), where J = |OH|/R, as at X₁₁₁₃.

  Accordingly, the isotomic conjugate, T(k), and the isogonal conjugate, U(k), are given by

  T(k) = (a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Cos[k]^2 - 4*(a^4 - b^4 - c^4)*S^2*Sin[k]^2 : :

  U(k) = a^2 ((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Cos[k]^2 - 4*(a^4 - b^4 - c^4)*S^2*Sin[k]^2) : :

  V(k) lies on the Jerabek rectangular hyperbola {{A, B, C,X₃,X₄}}.

  T(k) lies on the line X₆₉X₂₆₄.

  U(k) lies on the Euler line.

  X₆₁₄₅ = perspector of the Vu (π/4) conic, and X₁₈₄₃₄ = perspector of the Vu (π/6) conic.

  For every real number x, the perspector of the Vu (acrctan(x))-conic is

  1/((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(a^4 - b^4 - c^4)*S^2*x^2) : : , with isogonal conjugate

  a^2*((a^2 - b^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(a^4 - b^4 - c^4)*S^2*x^2) : :

  on the Euler line. (Peter Moses, May 15, 2020)

• X₃₈₄₅₁–X₃₈₄₅₆ — Points associated with Vu P-cirumcircle points

  based on notes from Vu Thanh Tung, May 16, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC, but not X₁ and not on the circumcircle. Let

  P' = isogonal conjugate of P

  T_a = line tangent to the circle APP’ at A, and define T_b and T_c cyclically.

  The lines T_a, T_b, T_c concur in a point V(P) on the circumcircle:

  V(P) = V(P') = a^2 / (c^2 p^2 q + b^2 p^2 r - a^2 q^2 r - a^2 q r^2) : :

  The point V(P) is here named the Vu P-circumcircle point.

  The appearance of (i,j) in the following list means that V(X_i) = X_j: (2,111), (3,74), (4,74), (5,14979), (6,111), (9,2291), (7,38451), (8,38452), (10,38453)

  See Vu Circumcircle Point (Vu%20Circumcircle%20Point.pdf).

  V(P) is the trilinear pole of the line through X₆ and the PK-transform of P. (Randy Hutson, May 19, 2020)

• X₃₈₄₅₇–X₃₈₄₆₅ — Points associated with Vu (P,U)-circles points

  based on notes from Vu Thanh Tung, May 18-19, 2020. Let P =\(p : q : r\) and U =\(u : v : w\) be distinct points in the plane of a triangle ABC, not both on the circumcircle. Let A' be the point, other than A, that lies on both circles (ABC) and (APU), and define B' and C' cyclically. The four lines AA', BB″, CC″, PU concur in a point, V(P,U), here named the Vu {P,U}-circles point point, given by

  V(P,U) = V(U,P) = a^2 (- p y w (p + r + q) + q r u (u + v + w)) + b^2 p u (r (u + v) - (p + q) w) + c^2 p u (-(p + r) v + q (u + w)) : :

  The appearance of (i,j,k) in the following list means that V(X_i,X_j) = V(X_j,X_i) = X_k: (1,2,7292), (1,3,36), (1,6,16784), (1,4,1870 ), (1,5,38458), (1,6,16784), (1,7,38459), (1,8,38460), (2,3,23), (2,4,468), (2,5,37760), (2,6 11580), (2 7,37761), (2,8,37762), (3,4,186), (3,5,2070), (3,6,187), (3,7,32624), (3,8,17100), (4,5,37943), (4,6,8744), (4,75,38457), (13,14,1989), (15,16,6), (61,62,35006), (371,372,1692)

  See Vu PU Circles Point (VuPUCirclesPoint.pdf).

  Let V₂(P,U) denote the point introduced in the preamble just before X₃₇₇₅₆, given by

  V₂( (P, U) = q r (a^2 (q r u (u+v+w) - p v w (p+q+r)) - b^2 p u (w (p+q) - r (u+v)) - c^2 p u (v (p+r) - q (u+w))) : :

  Then

  V(P,U) = barycentric product P* V₂(P,U)

  V(P,U) = barycentric product U* V₂(U,P)

  V₂(P,U) = barycentric product U*V₂(U,P)

  V(X₂,U) = V(U,X₂) = V₂(X₂,U)

  V(X₃,U) = V(U,X₃) = circumcircle-inverse of U

  The Vu {P,U}-circles point is related to the Vu circlecevian point as follows: Let P' and U' be the isogonal conjugates of P and U, resp. Then the Vu {P,U}-circles point is the isogonal conjugate of the Vu circlecevian point V(P',U'). (Randy Hutson, May 20, 2020)

• X₃₈₄₇₃–X₃₈₄₈₅ — Points associated with the Conway circle

  contributed by Peter Moses, May 20, 2020. Let P =\(p : q : r\) be a point on the circumcircle of a triangle ABC. Then the complement of the Conway-circle-inverse of P, denoted by CC(P) lies on the Apollonius circle. The appearance of (i,j) in the following list means that CC(X_i) = X_j: (101,3032), (106,3034), (109, 38485), (111,6044), (729,5213), (739,3030),(2291,34458), (8693,3033), (26715,34456), (28841,3029), (32722,34459), (32726,34455).

• X₃₈₄₈₇–X₃₈₄₉₄ — Points associated with Vijay-Paasche-Hutson triangles

  contributed by Dasari Naga Vijay Krishna, May 21, 2020. In the preamble just before X₃₇₉₉₄, six points are defined as follows:

  A_b =\(0 : 2R : c\), A_c =\(0 : b : 2R\), B_c =\(a : 0 : 2R\), B_a =\(2R : 0 : c\), C_a =\(2R : b : 0\), C_b =\(a : 2R : 0\) (These points lie on the Paasche ellipse; see X₃₇₈₆₁ and X₃₇₈₈

  Here we define six more points:

  A'_b = 0 : 4R + c : c and A'_c =\(0 : b : 4R\) + b

  B'_c =\(a : 0 : 4R\) + a and B'_a = 4R +\(c : 0 : c\)

  C'_a = 4R+\(b : b : 0\) and C'_b = a : 4R+a : 0

  Q_a = midpoint of A'_bA'_c, Q_b = midpoint of B'_cB'_a, Q_c = midpoint of C'_aC'_b

  Geometrically, A'_b = midpoint of B and A_b, A'_c = midpoint of C and A_c, etc.. Using the notation in the preamble just before X₃₈₃₁₀, these midpoints are given by taking (p,q,r) = (4R + a, 4R + b, 4R + c) and (u,v,w) = (a,b,c).

  Define 21 points as follows:

  L'_a = A'_cC'_a∩A'_bB'_a, L'_b = A'_bB'_a∩B'_cC'_b, L'_c = B'_cC'_b∩A'_cC'_a

  P'_a = C'_aB'_c∩C'_bB'_a, P'_b = C'_bA'_c∩C'_aA'_b, P'_c = B'_aA'_c∩B'_cA'_b

  K'_a = C'_bA'_c∩A'_bB'_c, K'_b = A'_cB'_a∩B'_cC'_a, K'_c = B'_aC'_b∩C'_aA'_b

  M_a = C'_bA_c∩B'_cA_b, M_b = C'_aB_c∩A'_cB_a, M_c = A'_bC_a∩B'_aC_b

  N_a = B'_aC_a∩C'_aB_a, N_b = C'_bA_b∩A'_bC_b, N_c = A'_cB_c∩B'_cA_c

  V_a = A'_cB'_c∩A'_bC'_b, V_b = B'_aC'_a∩B'_cA'_c, V_c = A'_bC'_b∩B'_aC'_a

  \(R_{a}\)= B'_aC_b∩C'_aB_c, \(R_{b}\) = A'_bC_a∩C'_bA_c, \(R_{c}\) = B'_cA_b∩A'_cB_a

  First barycentrics representing the 21 points follow: (omitted here)

  Also, from the preamble just before X₃₇₉₄₄,

  L_a = 4R^2 - b c : b (2R + c) : c (2R + b) )

  P_a = 16R^4 - a^2 b c : 2R b (4 R^2 - a c) : 2R c ( 4R^2 - a b)

  K_a = a (4R^2 - b c) : 4R^2 (2R + b) : 4R^2 (2R + c)

  H_a = -a (4 R^2 - b c) : 2 R b (2R + c) : 2 R c (2R + b) )

  Related triangles are here given names as follows:

  L'_aL'_bL'_c = 4th Vijay-Paasche-Hutson triangle

  P'_aP'_bP'_c = 5th Vijay-Paasche-Hutson triangle

  K'_aK'_bK'_c = 6th Vijay-Paasche-Hutson triangle

  M_aM_bM_c = 7th Vijay-Paasche-Hutson triangle

  N_aN_bN_c = 8th Vijay-Paasche-Hutson triangle

  V_aV_bV_c = 9th Vijay-Paasche-Hutson triangle

  RaRbRc = 10th Vijay-Paasche-Hutson triangle

  Q_aQ_bQ_c = 11th Vijay-Paasche-Hutson triangle
  Collinearities: L'_a, H_b, H_c

  A, N_a, V_a, L_a, L'_a, H_a

  A, M_a, K_a are collinear

  P'_a, L'_a, K'_a are collinear

  V_a, K'_a, Q_a

  \(R_{a}\), N_a, P_a, are collinear

  (Each list of collinearities represents a family of collinearities; e.g., the list L'_a, H_b, H_c also represents L'_b, H_c, H_a and L'_c, H_a, H_b.)

  Perspectors of triangles:

  X₁₁₂₃ = ANaVaL’aLaHa ∩ BNbVbL’bLbHa ∩ CNcVcL’cLcHc (the Paasche point)

  X₃₀₈₃ = AMaKa ∩ BMbKb ∩ CMcKc = X₁X₂∩X₃₇X₄₉₄

  X₃₀₈₆ = AP_a ∩ BP_b ∩ CP_c = X₁X₂∩X₄X₁₁

  X₃₇₈₈₄ = AT_a ∩ BT_b ∩ CT_c

  X₃₇₈₆₁ = H_aT_aK_a ∩ H_bT_bK_b ∩ H_cT_cK_c = center of Paasche conic

  [and others]

• X₃₈₄₉₇–X₃₈₅₂₉ — Dilations of points on the circumcircle to other circles

  contributed by Clark Kimberling and Peter Moses, May 23, 2020. Suppose that P and U are distinct points in the plane of a triangle ABC. Let Γ(P,U)) denote the circle with center P and pass-through point U. Suppose that V is a point distinct from U, and let D(P,U,V) denote the dilation from P that maps Γ(P,U)) onto Γ(P,V)). Centers X₃₈₄₉₇-X₃₈₅₁₉ are dilations from the circumcircle to Γ(X₃,X₈), and centers X₃₈₅₂₀-X₃₈₅₂₉ are dilations from the circumcircle to Γ(X₃,X₇₆)

  The appearance of {{\(i_{1}\), \(i_{2}\), … \(i_{k}\)}} in the following list means that the k points all lie on the circle Γ(X₃,\(X(i_{1})\)): {{1,40,13534,22939}}, {{2,376,11006,14916}}, [and others].

• X₃₈₅₃₆–X₃₈₅₅₀ — Vu (P,U)-circles perspectors

  based on notes contributed by Thanh Tung, May 23, 2020. Let P =\(p : q : r\) (barycentrics) and U =\(u : v : w\) be points in the plane of a triangle ABC such that P, U, P₁, U₁, where P₁ and U₁ are the respective isogonal conjugates of P and U, are distinct finite points. Let A' be the point, other than A, in which the circles (APU) and (AP1U1) intersect, and define B' and C' cyclically. The triangles ABC and A'B'C' are perspective, and their perspector is given by

  V(P,U) = V(U,P) = V(P₁,U₁) = V(U₁,P₁) = (c^2 p q u^2 + b^2 p r u^2 + a^2 q r u^2 - c^2 p^2 u v + b^2 p r u v - c^2 p r u v + a^2 q r u v - b^2 p^2 u w - b^2 p q u w + c^2 p q u w + a^2 q r u w - a^2 p^2 v w - a^2 p q v w - a^2 p r v w)*(-a^2 c^2 p q u v + c^4 p q u v - a^2 c^2 q^2 u v + b^2 c^2 p r u v - a^2 c^2 p q v^2 - a^2 c^2 q^2 v^2 - a^2 c^2 q r v^2 + b^2 c^2 p q u w + b^4 p r u w + a^2 b^2 q r u w - a^2 c^2 q^2 v w + a^2 b^2 p r v w + a^4 q r v w - a^2 c^2 q r v w)*(c^4 p q u v + b^2 c^2 p r u v + a^2 c^2 q r u v + b^2 c^2 p q u w - a^2 b^2 p r u w + b^4 p r u w - a^2 b^2 r^2 u w + a^2 c^2 p q v w + a^4 q r v w - a^2 b^2 q r v w - a^2 b^2 r^2 v w - a^2 b^2 p r w^2 - a^2 b^2 q r w^2 - a^2 b^2 r^2 w^2) : :

  The point V(P,U)) is here named the Vu (P,U)-circles perspector.

  See Vu Circles Perspector (Vu%20Circles%20Perspector.pdf)

  The appearance of (i,j) in the following list means that V(X_i,X_j) = X_k:

  (2,3,14246) and (2,4,14246), (4,6,14246) and (3,6,14246), (2,5,38536), (2,7,38537), (2,8,38538), (3,5,38539), (3,7,38540), (3,8,38541), (4,5,38542), (4,7,38543), (4,8,38544), (5,6,38545), (5,7,38546), (5,8,38547), (6,7,38548), (6,8,38549), (7,8,38550)

• X₃₈₇₂₃–X₃₈₈₇₀ — Circum-Euler-points

  contributed by César Eliud Lozada, May 28, 2020. The centroids of the four triangles determined by four concyclic points are concyclic. . (Reference: Halsted, George Bruce, Elementary Synthetic Geometry, 1892, problem 24, pp. 84.) Indeed, the above result is true for other points distinct from the centroids, although the exact locus is still unknown.

  Part of this locus is as follows: let Q be a point on the circumcircle of ABC and P a point on the Euler line of ABC such that |OP|/|OH| = λ, λ being a real number not depending on ABC. Denote P_a = P-of-QBC, P_b = P-of-QCA and P_c = P-of-QAB. Then the points P, P_a, P_b, P_c are concyclic on a circle with radius r’ = |λ| R and whose center O'(Q, P) is here named the Q-circum-Euler-point of P. In this case, O'(Q, P) = λ (Q + X₄) + (1 - 2 λ) X₃, and it lies on the line {X₃, midpoint(X₄, Q) }. Moreover, when P is fixed and Q varies, the locus of O'(Q, P) is another circle with the same radius r’ and center P.

  There are other P such that P, P_a, P_b, P_c are concyclic. A numerical calculus shows that a partial list of such points contains P=X_n for n ∈ {1, 13, 14, 15, 16, 23, 26, 36, 40, 80, 125, 155, 165, 186, 265, 368, 369, 370, 399, 1144, 1147, 1385, 1482, 1511, 1658, 2070, 2071, 2072, 2077, 3167, 3232, 3576, 3579, 5159, 5373, 5394, 5473, 5474, 5537, 5609, 5611, 5615, 5626, 5899, 5961, 5962, 5963, 5964, 6104, 6105, 6699, 6771, 6774, 7387, 7464, 7575, 7689, 7982, 7987, 7991, 8008, 8009, 8148, 8697, 9909} (n<10000). When P=X₁, P, P_a, P_b, P_c are vertices of a rectangle.

  Curiously, when P=X₁₅ and Q varies, the locus of O'(Q,P) are the sides of a central triangle having A-vertex with barycentric coordinates A' = (\(\sqrt{3}\)*S+SA)*(SB+SC): S^2-SA*SC : S^2-SA*SB. This triangle is perspective to the following triangles with perspector X₃: (ABC, ABC-X3 reflections, 2nd anti-extouch, 2nd Hyacinth, Lucas antipodal(±1), Lucas central(±1), X3-ABC reflections). It is also perspective to the following triangles with the given perspectors: (circumsymmedial, 11485), (outer-Le Viet An, 3129), (symmedial, 61), (tangential, 22236). A similar locus does not occur for P=X₁₆.

  The following table contains the Q-circum-Euler-point of P for selected P and Q [table omitted here].

• X₃₈₈₀₉–X₃₈₈₂₄ — Vu cevian tangential perspectors

  based on notes from Vu Thanh Tung, May 31, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC, and let

  A₁B₁C₁ = cevian triangle of P

  \(t_{a}\) = line tangent to the circle (AB'C') at A, and define \(t_{b}\) and \(t_{c}\) cyclically

  A' = \(t_{b}\)∩\(t_{c}\), and define B' and C' cyclically.

  The triangle A'B'C' is the anticevian triangle of the point

  V(P) = a^2/(p q + p r) : b^2/(q r + q p) : c^2/(r p + r q),

  which is the perspector of ABC and A'B'C'.

  See Vu Cevian Tangential Perspector (VuCevianTangentialPerspector.pdf).

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (1,81), (2,6), (3,275), (4,2), (5,288), (6,83), (7,1), (8,57), (9,1170), (10,1171), (11,38809), (31, 38810), (32, 38811), (75,58), (76,251), (560,38812), (561, 38813)

  The triangle A'B'C' is also perspective to A₁B₁C₁ , the perspector, denoted by V'(P), being the P-Ceva-conjugate of V(P):

  V'(P) = a^2/(p(q+r)) * (b^2/(q^2(r+p)) + c^2/(r^2(p+q)) - a^2/(p^2(q+r))) : :

  The appearance of (i,j) in the following list means that V'(X_i) = X_j:

  (1,38814), (2,3), (3,38815), (4,193), (5,38816), (6,38817), (7,57), (8,34488), (9,38818), (10,38819), (31,38820), (32,38821)

  V(P) is the cevapoint of X₆ and P', where P' is the isogonal conjugate of P. (Randy Hutson, May 31, 2020)

• X₃₈₈₂₅–X₃₈₈₄₃ — Vu cevian-circles perspectors and related cyclocevian conjugates

  based on notes from Vu Thanh Tung, June 1-3, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC, but not on any of the medians (i.e., the lines AG, BG, CG), and let

  A₁B₁C₁ = cevian triangle of P

  A₂ = the point, other than A, in which the circles (ABC) and (AB₁C₁) intersect, and define B₂ and C₂ cyclically

  A' = BB₂∩CC₂, and define B' and C' cyclically

  The triangles ABC and A'B'C' are perspective, and the perspector is the point

  V(P) = a^2*(p + q)*(p + r) : b^2*(q + r)*(q + p) : c^2*(r + p)*(r + q).

  Also, V(P) = complement of the anticomplementary conjugate of P (César Lozada, June 4, 2020)

  Also, V(P) is the isogonal conjugate of the complement of P. (Randy Hutson, June 9, 2020)

  A'B'C' is the anticevian triangle of V(P), given by

  A' = -a^2*(p + q)*(p + r) : b^2*(q + r)*(q + p) : c^2*(r + p)*(r + q).

  See Vu Cevian-Circles Perspector (VuCevianCirclesPerspector.pdf).

  The triangle A'B'C' is here named the Vu cevian-circles triangle of P.

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (1,58), (3,54), (4,4), (5,1173), (6,251), (7,57), (8,1), (9,1174), (10,1126), (11,18771), (21,38825), (22, 38828), (23, 38831), (32,38826), (75,81), (76,83), (560,38827), (1501,38829), (1502,38830), (1928,38843)

  Let V'(P) = cyclocevian conjugate of V(P) wrt A'B'C', which is also the perspector of A'B'C' and A₂B₂C₂, given by

  V'(P) = a^2*(p + q)*(p + r)*(p*q + p*r - q*r) : :

  V'(P) is the barycentric product V(P)*[P-Ceva conjugate of X₂] = V(P)*[anticompliment of isotomic conjugate of P]. (Randy Hutson, June 9, 2020)

  The appearance of (i,j) in the following list means that V'(X_i) = X_j:

  (1,38832), (3, 26887), (4, 6353), (5,38833), (6,38834), (7,1420), (8,165), (9,38835), (10,38836), (31,38837), (32,38838), (75,21), (76,1799), (560,38839), (561,38840), (1501, 38841), (1502, 38842)

  The triangle A₂B₂C₂ is here named the Vu perspectivities triangle of P; see the preamble just before X₃₈₈₄₈. Barycentrics for A₂ are given by

  A₂ = a^2*(p + q)*(p + r) : b^2*(q - r)*(p + q): c^2*(r - q)*(p + r) : :

• X₃₈₈₄₈–X₃₈₈₈₇ — Vu tangential transforms

  contributed by Clark Kimberling and Peter Moses, June 7, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. The Vu perspectivies triangle of P, here denoted by T(P), is introduced in the preamble just before X₃₈₈₂₅. The A-vertex is given by

  a^2*(p + q)*(p + r) : b^2*(q - r)*(p + q) : c^2*(r - q)*(p + r) : :

  T(P) is perspective to the anticomplementary triangle for every P on the de Longchamps axis (a^2 x + b^2 y + c^2 z = 0).

  T(P) is perspective to the anticomplementary triangle for every P on the following conic:

  a^2(b^2 - c^2)x^2 + b^2(c^2 - a^2)y^2 + c^2(a^2 - b^2)z^2 = 0, this being the anticomplement of the conic {{A, B, C, X₂, X₆}}, which passes through X_i for these i: 2, 69, 75, 1272, 1369, 1370, 6527, 14360, 17135, 17149, 18133, 18750, 19583, 20245, 20351, 20934, 22339, 22340, 25332, 30698, 32747.

  T(P) is perspective to the excentral triangle for every P on the anticomplement of the excentral triangle.

  T(P) is perspective to the excentral triangle for every P on the following conic: anticomplement of the conic {{A, B, C, X₁, X₂}}, which passes through X_i for these i: 2, 8, 329, 556, 1655, 2895, 8055, 17794, 18297, 20344, 21219, 30578, 30695.

  T(P) is perspective to the tangential triangle for every P. The perspector, here named the Vu tangential transform of P, denoted by VT(P), is given by

  VT(P) = a^2*(p^2 + p*q + p*r - q*r) : b^2*(q^2 + q*r + q*p - r*p) : c^2*(r^2 + r*p + r*q - p*q).

  Then VT(P)) = isogonal conjugate of the perspector of ABC and the reflection of the cevian triangle of P in the centroid of ABCP (or the complement of the complement of P). It is also the perspector T₁(X₆,P), as defined in the preamble just before X₃₃₇₆₀. (Randy Hutson, June 9, 2020)

  If P lies on the circumcircle, then VT(P) = isogonal conjugate of P wrt its cevian triangle. If P lies on the line at infinity, then VT(P) = isogonal conjugate of P. (Randy Hutson, June 9, 2020)

  The Vu tangential transform of the Euler line is a conic centered at X₁₅₆₄₇ and passing through X_i for these i: 6, 24, 74, 1498, 1614, 38848, 38850, 38851, 38852, 38867, 38879, 38885. (Randy Hutson, June 9, 2020)

  From Peter Moses (June 10, 2020): Suppose that P lies on a line u x + v y + w z = 0. Then VT(P) lies on the following conic:

  4*b^4*c^4*u^2*(v - w)^2*x^2 - a^4*b^2*c^2*(u^4 - 2*u^2*v^2 + v^4 - 4*u^2*v*w + 8*u*v^2*w - 4*v^3*w - 2*u^2*w^2 + 8*u*v*w^2 - 2*v^2*w^2 - 4*v*w^3 + w^4)*y*z + (cyclic) = 0.

  Examples:

  conic VT(Euler line) passes through X_i for these i: 6, 24, 74, 1498, 1614, 1620, 35217, 35218, 35219

  conic VT(X₁X₃) passes through X_i for these i: 104, 595, 1614

  conic VT(Brocard axis) passes through X_i for these i: 98, 1614, 1627, 33773

  conic VT(Soddy line) passes through X_i for these i: 103, 595, 1498, 1617, 1619, 1622

  conic VT(Nagel line) passes through X_i for these i: 3, 6, 106, 595, 1616, 23374, 33771, 33804, 35223

  conic VT(X₂X₆) passes through X_i for these i: 6, 22, 111, 1611, 1627, 33774, 35212, 35216

• X₃₈₉₀₀–X₃₈₉₂₉ — Vu PCC perspectors

  contributed by Vu Thanh Tung, June 22, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC, not on any of the lines AO, BO, CO, where O = X₃ = circumcenter. Let

  A₁B₁C₁ = pedal triangle of P

  A₂ = the point, other than A, where the circles (ABC) and (AB₁C₁) meet, and define B₂ and C₂ cyclically

  A₃ = BB₂∩CC₂, define B₃ and C₃ cyclically

  A₄B₄C₄ = circumcevian triangle of P.

  Then A₄B₄C₄ is perspective to A₂B₂C₂, and the perspector is the point V(P), here named the Vu 1st PCC perspector, given by

  V(P) = a^2 (2 b^2 c^2 p^2 + a^2 c^2 p q + b^2 c^2 p q - c^4 p q + a^2 b^2 p r - b^4 p r + b^2 c^2 p r + a^4 q r - a^2 b^2 q r - a^2 c^2 q r) : :

  Also, A₄B₄C₄ is perspective to A₃B₃C₃, and the perspector is the point T(P), here named the Vu 2nd PCC perspector, given by [coordinates omitted herej].

  See Vu Pedal Circles Circumcevian Perspector (Vu%20Pedal%20Circles%20Circumcevian%20Perspector.pdf).

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (1,56),(2,1995), (4,24), (5,13621), (6,1384), (7,38900), (8,38901), (9,38902), (10,38903), (31,38904), (32,38905), (75,38906), (76,38907), (83,38908), (141,38909), (560,38910), (561,38911), (1501,38912), (1502,38913), (1928,38914), (2321,38915), (2887,38916), (3676,38917)
  The appearance of (i,j) in the following list means that T(X_i) = X_j:

  (1,1394), (2,38918), (4,4), (5,38919), (6,38920), (8,38922), (9,38923), (10,38924), (31,38925), (32,38926), (75,38927) (76,38928), (83,38929)

  V(P) is the isogonal conjugate of the pedal antipodal perspector of P. (Randy Hutson, June 17, 2020)

• X₃₈₉₅₇–X₃₉₁₀₂ — Crosssums of circumcircle-P-antipodes and line intercepts

  contributed by Randy Hutson, June 17, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of triangle ABC. Let P' be the isogonal conjugate of P. Then the locus of the crosssum of circumcircle-P-antipodes is the bicevian conic of X₂ and P', with center 2 a^2/p + b^2/q + c^2/r : :. For example, the locus of the crosssum of circumcircle antipodes is the nine-point circle, which is the bicevian conic of X₂ and X₄, and the locus of the crosssum of circumcircle-X₆-antipodes is the Steiner inellipse, which is the limit of the bicevian conic of X₂ and Q as Q approaches X₂.

  Let L be a line. The crosssum of the (real or nonreal) circumcircle-intercepts of L is the X₂-Ceva conjugate of the crossdifference of every pair of points on L, and is the center of the circumconic that is the isogonal conjugate of L. If L passes through X₃, then the crosssum is also the orthopole of L, and lies on the cevian circle of the isogonal conjugate of every point on L.

• X₃₉₁₀₃–X₃₉₁₀₈ — Points on the self-dual permutation ellipse

  contributed by Clark Kimberling and Peter Moses, June 20, 2020. Suppose that Γ is a conic. The dual of Γ is the conic consisting of points P =\(p : q : r\) such that the line px + qy + rz = 0 is tangent to Γ. See, for example, Paul Yiu’s Introduction to the Geometry of the Triangle, p. 125.

  The dual of a permutation ellipse (defined in the preamble just before X₃₄₃₄₁), is also a permutation ellipse. There exists a unique self-dual permutation ellipse, given by the equation

  x^2 + y^2 + z^2 + 4(y z + z x + x y) = 0.

  If U =\(u : v : w\) is a point on the Steiner circumellipse, then the point

  D(U) = 3 u + (\(\sqrt{2}\) - 1)(u + v + w) : :

  is on the self-dual permutation ellipse. The appearance of (i,j) in the following list means that D(X_i) = X_j: (99,39103), (190,39104), (671,39105), (903,39106), (6189,39107), (6190,39108).

• X₃₉₁₄₁–X₃₉₁₄₆ — Perspectors related to the obverse triangle of X₆₉

  contributed by Clark Kimberling and Peter Moses, June 25, 2020. The obverse triangle A'B'C' of X₆₉ is the central triangle given by

  A' = -a^2 + b^2 + c^2 : a^2 + b^2 - c^2 : a^2 - b^2 + c^2.

  See the preamble just before X₂₄₃₀₇ for the definition of obverse triangle.

• X₃₉₁₅₈–X₃₉₁₆₁ — Foci of the real and imaginary Steiner ellipses

  contributed by Peter Moses, July 2-3, 2020. The foci of the Steiner circumellipse and the Steiner inellipse are all given by the following form for 1st barycentric:

  2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - sgn1*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\) + const*S*sgn2*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + sgn1*2*a^2*b^2*c^2*\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\)*J^2]

  The Steiner circumellipse has const = 1. Specifically, X₃₉₁₅₈ is given by (sgn1,sgn2) = (1,1)

  X₃₉₁₅₉ is given by (sgn1,sgn2) = (1,-1)

  X₃₉₁₆₀ is given by (sgn1,sgn2) = (-1,1)

  X₃₉₁₆₁ is given by (sgn1,sgn2) = (-1,-1)
  The Steiner inellipse has const = 2. Specifically, X₃₉₁₆₂ is given by (sgn1,sgn2) = (1,1)

  X₃₉₁₆₃ is given by (sgn1,sgn2) = (1,-1)

  X₃₉₁₆₄ is given by (sgn1,sgn2) = (-1,1)

  X₃₉₁₆₅ is given by (sgn1,sgn2) = (-1,-1)

  See the preamble just before X₃₉₂₀₂ for vertices of the two ellipses.

• X₃₉₁₇₇–X₃₉₂₀₁ — Vietnamese points

  contributed by Vu Thanh Tung, July 3, 2020. Suppose that T₁ = U₁V₁W₁ and T₂ = U₂V₂W₂ are triangles and that T₁ is not perspective to T₂. Let UVW be the vertex triangle of T₁ and T₂. Let L_U = radical axis to the circles (UV₁W₁) and (UV₂W₂), and define L_V and L_W cyclically.

  The lines L_U, L_V, L_W concur in a point Vn(T₁,T₂), here named the Vietnamese point of T₁ and T₂, denoted by Vn(T₁, T₂). This result is based on Problem 1 in the IMO (International Mathematics Olympiad) 1995 Vietnam TST (Team Selection Test).

  See Vietnamese Point (VietnamesePoint.pdf).

  Next, suppose that P =\(p : q : r\) and U =\(u : v : w\) are points, and let T₁ = cevian triangle of P, and T₂ = cevian triangle of U. Then

  Vn(T₁, T₂) = a^2 (p + q) (p + r) (u + v) (u + w) (r v - q w) : :

  Let T₃ = anticevian triangle of P, and T₄ = anticevian triangle of U. Then

  Vn(T₃, T₄) = (p - q - r) (-q u + p v) (u - v - w) (-r u + p w) (b^2 (-(q - r)^3 u^2 w + p^3 u w (-v + w) + p^2 (r u^2 (v - 2 w) - r (v - w)^3 + q u^2 w) + p u (q^2 (v - w) w + r^2 (u^2 - v^2 + 3 v w - 2 w^2) + q r (-u^2 + v^2 - 4 v w + 3 w^2))) + c^2 ((q - r)^3 u^2 v + p^3 u v (v - w) + p u (r^2 v (-v + w) + q^2 (u^2 - 2 v^2 + 3 v w - w^2) + q r (-u^2 + 3 v^2 - 4 v w + w^2)) + p^2 (r u^2 v + q ((v - w)^3 + u^2 (-2 v + w)))) + a^2 (p^3 u v w + (q - r) (r^2 v (-u^2 + v (v - w)) + 2 q r v w (-v + w) + q^2 w (u^2 + (v - w) w)) - p u (q^2 v w + r^2 v w + q r (-u^2 + v^2 - 4 v w + w^2)) - p^2 (q w (u^2 + (v - w) w) + r v (u^2 + v (-v + w))))) : :

  Let T₅ = circumcevian triangle of P, and T₆ = circumcevian triangle of U. Then

  Vn(T₅, T₆) = a^2 (b^2 (r u - p w) + c^2 (p v - q u) - a^2 (q w - r v)) : : , this being the pole wrt the circumcircle of the line PU

  Let O = X₃ = circumcenter of ABC. If O, P, U are collinear, then Vn(T₅, T₆) is the infinite point on the line OPU.

  The appearance of (i,j,k) in the following list means that the Vietnamese point of the cevian triangle of X_i and the cevian triangle of X_j is X_k: (1,2,3733), (1,3,39177), (1,4,4560), (1,5,39178), (1,6,39179), (2,3,23286), (2,4,523), (2,5,39180), (2,6,18105), (3,4,15412), (3,5,39181), (3,6,39182), (4,5,39183), (4,6,4580), (5,6,39184)

  The appearance of (i,j,k) in the following list means that the Vietnamese point of the anticevian triangle of X_i and the anticevian triangle of X_j is X_k: (1,2,39185), (1,3,39186), (1,4,39187), (1,5,39188), (1,6,39189), (2,3,39190), (2,4,39192), (2,5,29192), (2,6,39193), (3,5,39194), (3,5,39195), (3,6,38861), (4,5,39196), (4,6,39197), (5,6,39198)

  The appearance of (i,j,k) in the following list means that the Vietnamese point of the circumcevian triangle of X_i and the circumcevian triangle of X_j is X_k: (1,2,4057), (1,3,513), (1,4,39199), (1,5,39200), (1,6,667), (2,3,523), (2,4,523), (2,5,523), (2,6,669), (3,4,523), (3,5,523), (3,6,512), (4,5,523), (4,6,39201), (5,6,34952)

• X₃₉₂₀₂–X₃₉₂₀₇ — Vertices of the real and imaginary Steiner ellipses

  contributed by Peter Moses, July 3, 2020. The vertices of the Steiner circumellipse and the Steiner inellipse are all given by the following form for 1st barycentric:

  2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - sgn1*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\) + 2*const*S*sgn2*Sqrt[\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\)/(a^2 + b^2 + c^2 + 2*sgn1*\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\))]*Sqrt[sgn1*(-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*sgn1*a^2*b^2*c^2*\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\)*J^2)]

  The Steiner circumellipse has const = 1. Specifically, X₃₉₂₀₂ is given by (sgn1,sgn2) = (1,1)

  X₃₉₂₀₃ is given by (sgn1,sgn2) = (1,-1)

  X₃₉₂₀₄ is given by (sgn1,sgn2) = (-1,1)

  X₃₉₂₀₅ is given by (sgn1,sgn2) = (-1,-1)
  The Steiner inellipse has const = 2. Specifically, X₃₉₂₀₆ is given by (sgn1,sgn2) = (1,1)

  X₃₉₂₀₇ is given by (sgn1,sgn2) = (1,-1)

  X₃₉₂₀₈ is given by (sgn1,sgn2) = (-1,1)

  X₃₉₂₀₉ is given by (sgn1,sgn2) = (-1,-1)

  The points X₃₉₂₀₂ and X₃₉₂₀₃ are endpoints of the major axis of the Steiner circumellipse, and X₃₉₂₀₄ and X₃₉₂₀₅ lies on the minor axis. The points X₃₉₂₀₆ and X₃₉₂₀₇ are endpoints of the major axis of the Steiner inellipse, and X₃₉₂₀₈ and X₃₉₂₀₉ lies on the minor axis.

• X₃₉₂₁₀–X₃₉₂₂₈ — Centers of Vietnamese circles

  :contributed by Vu Thanh Tung, July 4, 2020. Suppose that T₁ = U₁V₁W₁ and T₂ = U₂V₂W₂ are triangles and that T₁ is not perspective to T₂. Let UVW be the vertex triangle of T₁ and T₂. Let L_U = radical axis to the circles (UV₁W₁) and (UV₂W₂), and define L_V and L_W cyclically.

  The lines L_U, L_V, L_W concur in a point Vn(T₁,T₂), the Vietnamese point of T₁ and T₂, denoted by Vn(T₁, T₂), introduced in the preamble just before X₃₉₁₇₇. Continuing, let U₀ be the point, other than U, in which the circles (UV₁W₁) and (UV₂W₂) intersect and define V₀ and W₀ cyclically. Let M_i be the Miquel point of T_i for i = 1, 2; e.g.,

  M₁ = (UV₁W₁)∩(UV₁W₁)∩(U₁V₁W).

  The six points U₀,V₀,W₀,M₁,M₂,Vn lie on a circle, here named the Vietnamese circle of T₁ and T₂, denoted by v(T₁,T₂). The center of this circle is denoted by Vo(T₁,T₂).

  See Vietnamese Circle (VietnameseCircle.pdf).

  If P =\(p : q : r\) and U =\(u : v : w\) are distinct with distinct circumcevian triangles T₁, T₂ then v(T₁,T₂) passes through the circumcenter, O, and is the circumcircle-inverse of the line PU. If P, U, O are collinear then v(T₁,T₂) is the circle of infinite radius - that is, the line at infinity, and Vo(T₁,T₂) = O.

  If T₁ = cevian triangle of P, and T₂ = cevian triangle of U, then the appearance of (i,j,k) in the following list means that the center of the Vietnamese circle of the cevian triangle of X_i and the cevian triangle of X_j is X_k: (2,3,37084), (3,4,15451), (2,4,523)

  The appearance of (i,j,k) in the following list means that center of the Vietnamese circle of the anticevian triangle of X_i and the anticevian triangle of X_j is X_k: (1,2,2957), (2,6,30715)

  The appearance of (i,j,k) in the following list means that the center of the Vietnamese circle of the circumcevian triangle of X_i and the circumcevian triangle of X_j is X_k: (1,3,513), (2,3,523), (2,4,523), (2,6,5926), (3,4,523), (3,6,523)

• X₃₉₂₇₄–X₃₉₂₈₇ — Trilinear poles of Vietnamese lines

  contributed by Vu Thanh Tung, July 19, 2020. Suppose that T = U₁V₁W₁, T = U₂V₂W₂, T = U₃V₃W₃ are triangles such that U₁,U₂,U₃ are collinear, V₁,V₂,V₃ are collinear, and W₁, W₂, W₃ are collinear, so that the three pairs {T₂,T₃}, {T₃,T₁}, {T₁,T₂} have the same vertex triangle.

  Theorem: The Vietnamese points (defined in the preamble just before X₃₉₁₇₇) of the three pairs {T₂,T₃}, {T₃,T₁}, {T₁,T₂} are collinear.

  Corollary: Let T_P, T_U, T_X be the cevian triangles of points P =\(p : q : r\), U =\(u : v : w\), X =\(x : y : z\) (barycentrics). Then the three Vietnamese points of the three pairs {T_P,T_U}, {T_U,T_X}, {T_X,T_P}} lie on a line L'(P,U,X), here named the Vietnamese line of {P,U,X}, of which the trilinear pole is given by [coordinates omitted here].

  See Vietnamese Line (VietnamesePoint.pdf).

  See Vietnamese Point (VietnameseLine.pdf).
  The appearance of (i,j,k,v) in the following list means that L(X_i,X_j,X_k) = X(v):

  (1,2,3,39274), (1,2,4,14534), (1,2,5,39275), (1,2,6,39276), (1,3,4,39277), (1,3,5,39278), (1,3,6,39279), (1,4,5,39280), (1,4,6,39281), (1,5,6,39282), (2,3,4,275), (2,3,5,288), (2,3,6,39283), (2,4,5,39284), (2,4,6,83), (2,5,6,39285), (3,4,5,39286), (3,4,6,39287), (3,5,6,39288), (4,5,6,39289)

• X₃₉₃₀₀–X₃₉₃₀₇ — Centers of osculating circles to Steiner ellipses

  contributed by Peter Moses, July 23-24, 2020. The centers of osculating circles at the four vertices of the Steiner circumellipse and the Steiner inellipse are all given by the following form for 1st barycentric:

  2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\)*sgn1 + const*S*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\)*J^2*sgn1]*sgn2*Sqrt[2 - ((a^2 + b^2 + c^2)*sgn3)/\(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\)]

  The Steiner circumellipse has const = 1/2. Specifically, X₃₉₃₀₀ is given by (sgn1,sgn2,sgn3) = (1,1,-1)

  X₃₉₃₀₁ is given by (sgn1,sgn2,sgn3) = (1,-1,-1)

  X₃₉₃₀₂ is given by (sgn1,sgn2,sgn3) = (-1,1,1)

  X₃₉₃₀₃ is given by (sgn1,sgn2,sgn3) = (-1,-1,1)
  The Steiner inellipse has const = 1. Specifically, X₃₉₃₀₄ is given by (sgn1,sgn2,sgn3) = (1,1,-1)

  X₃₉₃₀₅ is given by (sgn1,sgn2,sgn3) = (1,-1,-1)

  X₃₉₃₀₆ is given by (sgn1,sgn2,sgn3) = (-1,1,1)

  X₃₉₃₀₇ is given by (sgn1,sgn2,sgn3) = (-1,-1,1)

  For the placement of the circles within the ellipses, see Steiner Cirumellipse Osculating Circles (SteinerCircumellipseOsculatingCircles.pdf) and Steiner Inellipse Osculating Circles (SteinerInellipseOsculatingCircles.pdf). See also the preambles just before X(39202 (vertices) and X₃₉₁₅₈ (foci).

• X₃₉₃₁₁–X₃₉₃₃₄ — Points associated with Paasche triangles

  contributed by Dasari Naga Vijay Krishna, July 24, 2020. In the preambles just before X₃₇₉₉₄ and X₃₈₄₈₇ we define points labeled as A_b, A_c, B_c, B_a, C_a, C_b , A'_b, A'_c, B'_c, B'_a, C'_a, C'_b . Here, we define six more points, A’’b, A’’c, B’’c, B’’a, C’’a, C’’b as the respective reflections of A_b, A_c, B_c, B_a, C_a, C_b in the vertices A, B, C.; e.g., A’’b = A_b in B, and A’’c = reflection of A_c in C, etc.

  Barycentrics for the 6 points are as follows:

  A”b = 0 : 2R+2c : -c

  A”c =\(0 : -b : 2R\)+2b

  B”c= \(-a : 0 : 2R\)+2a

  B”a = 2R+\(2c : 0 : -c\)

  C’’a = 2R+\(2b : -b : 0\)

  C’’b = -a : 2R+2a : 0

  It is easy to verify that the 6 points, A”b , A”c, B”c, B”a, C’’a, C’’b, lie on an ellipse, here named the Paasche Reflection Ellipse (PRE), given by the barycentric equation

  2 b c (R + a) x^2 + 2 c a (R + b) y^2 + 2 a b (R + c) z^2 + a (4 R^2 + 4 R b + 4 R c + 5 b c) y z + b (4 R^2 + 4 R c + 4 R c + 5 c a) x z + c (4 R^2 + 4 R a + 4 R b + 5 a b ) x y = 0.

  The center of the PRE is

  X₃₉₃₁₁ = a(16R^4(b+c-a)+16R^3(b+c)(-a+b+c)+4R^2bc(-4a+5b+5c)+8Rb^2c^2+ab^2c^2) : :

  The PRE is a “Kimberling generalized Paasche conic” given by (p,q,r) = (2(R+a),2(R+b),2(R+c)) and (u,v,w) = (-a,-b,-c); see the preamble just before X₃₈₃₁₀.

  Define 54 points (18 triangles) by the following intersections: [list omitted here].

• X₃₉₃₃₅–X₃₉₃₄₄ — Points on the conic TC(X₁₀₅₄)

  contributed by Clark Kimberling and Peter Moses, July 24-25, 2020. This preamble extends the preamble just before X₃₆₂₅₆, in which, for a point P =\(p : q : r\) (trilinears), the trilinear permutation conic denoted by TC(P), is defined as the conic that passes through the six points

  \(p : q : r\),\(q : r : p\),\(r : p : q\),\(p : r : q\),\(q : p : r\),\(r : q : p\).

  An equation for TC(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  Thus, TC(P) is analogous, and symbolically identical, to the permutation ellipse E(P), defined in the preamble just before X₃₄₃₄₁.

  Suppose that P =\(p : q : r\) (trilinears). For the rest of this paragraph,\(p:q:r\) are trilinears, but all other coordinates, and the equation for TC(P), are in barycentric coordinates. The conic TC(P) passes through the six points

  \(ap : bq : cr\),\(aq : br : cp\),\(ar : bp : cq\),\(ap : br : cq\),\(aq : bp : cr\),\(ar : bq : cp\).

  (Note that these six points are not six permutations of \(ap : bq : cr\).)

  An equation for TC(P) is (q r + r p + p q)(b^2 c^2 x^2 + c^2 a^2 y^2 + a^2 b^2 z^2) - abc(p^2 + q^2 + r^2)(ayz + bzx + cxy) = 0.

  The conic TC(X₁₀₅₄), given by the trilinear equation x^2 + 3 y z + (cyclic) = 0. Define \(f(x,y,z) = (x-y)(x-z) - (y-z)^{2}\). If X =\(x : y : z\) (trilinears) is a point other than X₁ =\(1:1:1\) then the point F(X) =\(f(x,y,z) : f(y,z,x) : f(z,x,y)\) is on TC(X₁₀₅₄).

  The appearance of (i,j) in the following list means that the point F(X_i) = X_j is on TC(X₁₀₅₄):

  (2,9359), (4,2636), (6,1054), (19,2639), (31,2640), (100,9324), (171,9355), .

  Let f(\(x : y : z\)) = (x - y)(x - z) - (y - z)^2. If X =\(x : y : z\) and X' = x’ : y ‘ : z’ are collinear with X₁, then f(X') = f(X), on TC(X₁₀₅₄).

  If P is on the anti-orthic axis, (the line X₄₄ X₅₁₃), then the P-Ceva conjugate of X₁ is on TC(X₁₀₅₄.

  The excenters, with trilinears \(-1:1:1\),\(1:-1:1\),\(1:1:-1\), also lie on TC(X₁₀₅₄). Let EC denote the circumconic of ABC that passes through the excenters, so that EC is given by the barycentric equation

  g(a,b,c) y z + g(b,c,a) z x + g(c,a,b) yz = 0, where \(g(a,b,c) = (a^{2} + b^{2} - c^{2})(a^{2} - b^{2} + c^{2})\).

  EC passes through X_i for i = 107, 648, 653, 685, 687, 1897, 6330, 6331, 6335, 6336, 8764, 13149, 15352, 15459, 16080, 16081, 16082, 16813, 17983, 30450, 36306, 36309, 38342. Six examples follow:

  X₁₀₅₄ = X₁₀₇-of-excentral-triangle

  X₂₆₃₆ = X₃₀₄₅₀-of-excentral-triangle

  X₂₆₄₀ = X₁₆₈₁₃-of-excentral-triangle

  X₉₃₂₄ = X₁₆₀₈₀-of-excentral-triangle

  X₉₃₅₅ = X₆₄₈-of-excentral-triangle

  X₉₃₅₉ = X₁₅₃₅₂-of-excentral-triangle

• X₃₉₃₄₅–X₃₉₃₆₈ — Points on the permutation ellipse E(X₄₄₄₀)

  contributed by Clark Kimberling and Peter Moses, July 25-28, 2020. This preamble extends the preamble just before X₃₄₃₄₁, in which, for a point P =\(p : q : r\) (barycentrics), the barycentric permutation ellipse denoted by E(P), is defined as the ellipse that passes through these six points:

  \(p : q : r\),\(q : r : p\),\(r : p : q\),\(p : r : q\),\(q : p : r\),\(r : q : p\).

  An equation for E(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  Thus, TC(P) is analogous, and symbolically identical to, the trilinear permutation conic TC(P), defined and discussed in the preambles just before X₃₆₂₅₆ and X₃₉₃₃₅.

  The ellipse E(X₄₄₄₀), is given by the barycentric equation x^2 + 3 y z + (cyclic) = 0. Define \(f(x,y,z) = (x-y)(x-z) - (y-z)^{2}\). If X =\(x : y : z\) (trilinears) is a point other than X₂ =\(1:1:1\) then the point F(X) =\(f(x,y,z) : f(y,z,x) : f(z,x,y)\) is on E(X₄₄₄₀).

  The appearance of (i,j) in the following list means that the point F(X_i) = X_j is on E(X₄₄₄₀):

  (1,4440), (3,39352), (6,148), (11,39353), (31,39345), (32,39346), (63, 39351), (75,9263), (76,25054), (192,9263), (561,39347), (514,17486), (523,8591), (594,39348), (649,39354), (1086,39349), (693,39350)

  The vertices \(-1:1:1\),\(1:-1:1\),\(1:1:-1\) of the anticomplementary triangle also lie on E(X₄₄₄₀), so that this ellipse is the anticomplement of the Steiner circumellipse.

  Let f(\(x : y : z\)) = (x - y)(x - z) - (y - z)^2. If X =\(x : y : z\) and X' = x’ : y ‘ : z’ are collinear with X₂, then f(X') = f(X), on E(X₄₄₄₀). Example: if X is on the Euler line (but is not X₂), then f(X) = X₃₉₃₅₂.

  If P is on the line at infinity, then the P-Ceva conjugate of X₂, which is also the anticomplement of the isotomic conjugate of P, is on E(X₄₄₄₀.

• X₃₉₃₇₁–X₃₉₃₇₂ — Points associated with the Cip transform

  contributed by Peter Moses, August 3, 2020. The Cip transform is introduced in Suren, Moses, and Kimberling, “The Circumcevian-Inversion Perspector of Two Triangles”, to appear in Journal for Geometry and Graphics, as follows: In the plane of a triangle ABC, let P be a point that is not on one of the sidelines, BC, CA, AB. Let Γ be the circumcircle of ABC, and let DEF be the circumcevian triangle of P; that is, D is the point, other than A, in which the line AP meets Γ, and likewise for the vertices E and F. Let X, Y, Z be the reflections of P in D, E, F, respectively, and let X',Y',Z' be the Γ-inverse of X, Y, Z, respectively. Let P' = BY'∩CZ'. Then ABC and X'Y'Z' are perspective triangles, and P' is the circumcevian-inversion perspector of X'Y'Z' and ABC. The mapping P → P' is the Cip transform.

  If Λ is a conic that passes through the circumcenter, O, then it is the Cip transform of a circular cubic. If you have Geogebra, you can view seven examples, named according to the choice of Λ:

  CIP_ABCIO (CIP_ABCIO.ggb).

  CIP_BrocardCircle (CIP_BrocardCircle.ggb).

  CIP_FeuerbachOfTangential (CIP_FeuerbachOfTangential.ggb).

  CIP_Jerabek (CIP_Jerabek.ggb).

  CIP_KiepertOfMedial (CIP_KiepertOfMedial.ggb).

  CIP_Lester (CIP_Lester.ggb).

  CIP_Thomson-Gibert-Moses hyperbola (CIP_Thomson-Gibert-Moses%20hyperbola).

  Names for the seven circular cubics are introduced here as ABCIO circular cubic, Brocard circular cubic, Feuerbach-of-tangential circular cubic, Jerabek circular cubic, Kiepert-of-medial circular cubic, Lester circular cubic, and Thomson-Gibert-Moses-hyperbola circular cubic. For each of these, O is the Cip-image of O and every point on the infinity line. Aside from O, two of the seven circular cubics have Cip-fixed points: X₁₀₂ on the ABCIO circular cubic, and X₁₁₀ on the Thomson-Gibert-Moses-hyperbola circular cubic.

  Another point on the ABCIO circular cubic is X₃₈₅₉₉, and two on the Jerabek circular cubic are X₃₅₃₇₂ and X₃₅₄₆₅;

  If Λ is a conic that does not pass through O, then the curve having Λ as Cip image has formal degree 4. An example is the Steiner circumquartic , see CIP_SteinerCircum (CIP_SteinerCircum.ggb), which passes through X₁₂₁₈₈.

• X₃₉₃₈₂–X₃₉₆₄₀ — and X₃₉₆₂₈-X₃₉₆₄₀, Circumcenters of circumcevian polar triangles

  based on notes from Suren and Peter Moses (August-September, 2020). Let A'B'C' be the circumcevian triangle of a point P in the plane of a triangle ABC. Let L_a be the polar of A' wrt the circumcircle of BPC, and define L_b and L_c cyclically. Let A″ = L_b∩L_c, and define B″ and C″ cyclically. The triangle A″B″C″ is here named the circumcevian polar triangle of P. If P =\(p : q : r\) (barycentrics), then the A-vertex of A″B″C″ is given by \(u : v : w\), where [formulas omitted here].

  The circumcenter of A″B″C″, denoted by MSV(P), is given by

  a^2/(c^2*(a^2 + b^2 - c^2)*p^2*q + 2*a^2*c^2*p*q^2 - b^2*(a^2 - b^2 + c^2)*p^2*r + a^2*(a^2 - b^2 + c^2)*q^2*r - 2*a^2*b^2*p*r^2 - a^2*(a^2 + b^2 - c^2)*q*r^2) : :

  Theorems:

  MSV(P) = circumcircle-antipode of the Vu circumcircle point of P (defined in the preamble just before X₃₈₄₅₁.

  If P lies on the cubic K269, then MSV(P) = X₁₀₉.

  If P lies on the Neuberg, then MSV(P) = X₁₁₀.

  If P lies on the cubic K270, then MSV(P) = X₁₁₂.

  If P lies on the cubic K1156, then MSV(P) = X₁₂₉₆.

  The appearance of (i,j) in the following list means that MSV(X_i) = X_j:

  (2,1296), (3,110), (4,110), (5,1291), (6,1296), (7,20219), (8,39628), (9,28291), (13,110), (14,110), (15,110), (16,110), [and others]

• X₃₉₃₈₉–X₃₉₄₀₁ — Circlecevian Perspectors

  contributed by Vu Thanh Tung, Auguist 21, 2020. The circlecevian triangle of a point is defined in the preamble just before X₃₇₈₄₁, as follows: Let P =\(p:q:r\) (barycentrics) be a point in the plane of a triangle ABC. Let A' be the point, other than P, in which the line AP meets the circle (PBC), and define B' and C' cyclically; the triangle A'B'C' is called the circlecevian triangle of P with respect to triangle ABC by Floor van Lamoen (Hyacinthos #10039); see the preambles just before X₃₄₅₂₀ and X₃₄₈₉₂. Let A₁B₁C₁ be the circlecevian triangle of P. Let A₂ be the point, other than A, of intersection of the circles (ABC) and AB₁C₁), and define B₂ and C₂ cyclically. Let A₃ = BB₂∩CC₂, and define B₃ and C₃ cyclically. Theb ABC is perspective to A₃B₃C₃, and the perspector, here named the circlecevian perspector of P, is given by

  V(P) = a^2 (c^2 p q^2 + b^2 p^2 r + 2 b^2 p q r + a^2 q^2 r + b^2 p r^2) (c^2 p^2 q + c^2 p q^2 + 2 c^2 p q r + b^2 p r^2 + a^2 q r^2) : :

  Let P* be the isogonal conjugate of P; then V(P) = V(P*).

  See Circlecevian Perspector (CirclecevianPerspector.pdf). (Vu Thanh Tung)

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (1,1),

  (2,39389), (6,39389),

  (3,54), (4,54),

  (5,39390), (54,39390),

  (7,39391), (55,39391),

  (8,39392), (56,39392),

  (9,39393), (57,39393),

  (10,39394), (58,39394),

  (31,39395), (75,39395),

  (32,39396), (76,39396),

  (39,39397), (83,39397),

  (141,39398), (251,39398),

  (560,39399), (561,39399),

  (1501,39400), (1502,39401),

  (1917,39401), (1928,39401)

• X₃₉₄₀₂–X₃₉₄₁₁ — Vu Isodynamic Perspectors

  contributed by Vu Thanh Tung, August 21, 2020. Let P be a point in the plane of a triangle ABC. Let A₁ be the 1st isodnamic point, X₁₅, of triangle PBC, and define B₁ and C₁ cyclically. Let A₂ be the point of intersection, other than A, of the circles (ABC) and (AB₁C₁), and define B₂ and C₂ cyclically. Let A₃ = BB₂∩CC₂, and define B₃ and C₃ cyclically. The triangle A₃B₃C₃ is perpsective to ABC, and the perspector, V(P), is here named the 1st Vu isodynamic perspector. Note that A₃B₃C₃ is the anticevian triangle of V(P).

  If the above construction is carried out with the 2nd isodynamic point, X₁₆, instead of X₁₅, the resulting perspector, T(P), is the 2nd Vu isodynamic perspector.

  Barycentrics were found by Peter Moses (August 22, 2020):

  V(P) = a^2/(\(\sqrt{3}\)*a^2*((a^2 - b^2 - c^2)*q*r - b^2*r^2 - c^2*q^2) - 2*(a^2*q*r + b^2*r*p + c^2*p*q)*S) : :

  T(P) = a^2/(\(\sqrt{3}\)*a^2*((a^2 - b^2 - c^2)*q*r - b^2*r^2 - c^2*q^2) + 2*(a^2*q*r + b^2*r*p + c^2*p*q)*S) : :

  The appearance of (i,j) in the following list means that V(X_i) = X_j: (1,39402), (2,39404), (3,2981), (4,39406), (5,39408), (6,39410)

  The appearance of (i,j) in the following list means that T(X_i) = X_j: (1,39403), (2,39405), (3,6151), (4,39407), (5,39409), (6,39411)

  See Isodynamic Perspector (IsodynamicPerspector.pdf). (Vu Thanh Tung)

• X₃₉₄₅₃ — Vu Isogonal Perspectors

  contributed by Vu Thanh Tung, August 26, 2020. In the plane of a triangle ABC, let P be a point such that P and its isogonal conjugate, P', are distinct finite points. Let A₁B₁C₁ be the pedal triangle of P, and let A'₁B'₁C'₁ be the pedal triangle of P'. Let A₂ be the point of intersection, other than A, of the circles (ABC) and (AA₁A'₁), and define B₂ and C₂ cyclically. Let A₃ = BB₂∩CC₂, and define B₃ and C₃ cyclically. The triangle A₃B₃C₃ is perpsective to ABC, and the perspector, V(P), is here named the Vu isogonal perspector of P. Note that V(P') = V(P), and A₃B₃C₃ is the anticevian triangle of V(P).

  If P =\(p : q : r\) (barycentrics), then

  V(P) = 1/(p*(b^2*r + SA*q)*(c^2q + SA*r)) : :

  The appearance of (i,j) in the following list means that V(X_i) = X_j: (2,39353), (3,4), (4,4), (5,39454), (6,39353), (7,39455), (8,39456), (9,39457), (10,39458), (31,39459), (32,39460), (75,39459), (76, 39460)

  Generalization (Vu Thanh Tung, August 28, 2020):

  Theorem: In the plane of a triangle ABC, let A₁ and A'₁ be points on BC such that B, C, A₁, and A'₁ are distinct. Define B₁ and C₁ cyclically, and define B'₁ and C'₁ cyclically. Let A₂ be the point of intersection, other than A, of the circles (ABC) and (AA₁A'₁), and define B₂ and C₂ cyclically. Let A₃ = BB₂∩CC₂, and define B₃ and C₃ cyclically. Then A₃B₃C₃ is perspective to ABC if and only if the points A₁, A'₁, B₁, B'₁, C₁, C'₁ lie on a conic.

  Corollary: Let A₁B₁C₁ be the cevian triangle of P, and let A'₁B'₁C'₁ be the cevian triangle of P'. Then the six vertices lie on the bicevian conic of P and P', and the perspector of the triangles ABC and A₃B₃C₃ is the barycentric product P*P'.

  See Isogonal Perspector (IsogonalPerspector.pdf) and Six Points Conic (SixPointsConic.pdf). (Vu Thanh Tung)

  Let O_a be the center of the involution {B, C}, {A₁, A'₁} (as in the preamble just before X₁₄₇₈₂) and define O_b and O_c cyclically. The points O_a, O_b, O_c are collinear on a line whose trilinear pole is the Vu-isogonal perspector of P or P'. (César Lozada, March 3, 2021.)

• X₃₉₄₆₉–X₃₉₄₇₄ — Yiu infinity points

  contributed by Clark Kimberling and Peter Moses, August 30, 2020. Suppose that F =\(f : g : h\) (barycentrics) is a point on the infinity line, L. The orthopoint (a.k.a. orthogonal conjugate) of F, denoted here by F' =\(f' : g' : h'\) also lies on L. In Introduction to the Geometry of the Triangle,, Paul Yiu notes that

  \(S_{A}\)*f*f’ + \(S_{B}\)*g*g’ + \(S_{C}\)*h*h’ = 0.

  Equialvently, the point F″ = \(S_{A}\)*f*f’ : \(S_{B}\)*g*g’ : \(S_{C}\)*h*h’

  also lies on L. The point F″ is here named the Yiu infinity point of F. Note that F″ = F*F'/H, where H = X₄, the orthocenter.

  The appearance of (i,j,k) in the following list means that X_i is on L, X_j = orthopoint of X_i, and X_k = Yiu infinity point of X_i:

  (30, 523, 9033)

  (511, 512, 39469)

  (512, 511, 39469)

  (513, 517, 8677)

  (514, 516, 39470)

  (515, 522, 39471)

  (516, 514, 39470)

  (517, 513, 8677)

  (519, 3667, 39472)

  (522, 515, 39471)

  (523, 30, 9033)

  (525, 1503, 39473)

  (542, 690, 39574)

  (690, 542, 39474)

  (1503, 525, 39475)

  There is a connection between Yiu infinity point and the Psi mapping (TCCT, p. 80): If Q is the the Yiu infinite point of a point P and the orthopoint of P), and if U is a point on the line of X₃ and the isogonal conjugate of P, then Psi(U) is the isogonal conjugate of Q. (Peter Moses, January 27, 2021)

• X₃₉₄₇₅–X₃₉₄₈₁ — Centers of circumcircle-inversion circles

  contributed by Clark Kimberling and Peter Moses, September 1, 2020. Suppose that Γ is the circle with powers (u,v,w) – that is, u = power of A with respect to the circle. and v and w are defined cyclically. Suppose that L is a line, given by p x + q y + r z = 0 (barycentrics), that does not pass through X₃. The Γ-inverse of L is a circle that passes through X₃. This circle is here named the Γ-inversion circle of L, denoted by ((Γ, L)). The center of ((Γ, L)) is given by

  \(f(a,b,c,u,v,w,p,q,r) : f(b,c,a,v,w,u,q,r,p) : f(c,a,b,w,u,v,r,p,q)\), where [long equation omitted here].

  In particular, the center of ((circumcircle, L)) is the point

  a^2*(a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*p - b^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*q - c^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*r) : : , and the squared radius of ((circumcircle, L)) is

  (a^4*b^4*c^4*(a^2*p^2 + (-a^2 - b^2 + c^2)*p*q + b^2*q^2 + (-a^2 + b^2 - c^2)*p*r + (a^2 - b^2 - c^2)*q*r + c^2*r^2))/((a + b - c)*(a - b + c)*(-a + b + c)*(a + b + c)*(a^2*(a^2 - b^2 - c^2)*p - b^2*(a^2 - b^2 + c^2)*q - c^2*(a^2 + b^2 - c^2)*r)^2).

  If P and P' are distinct points on L, then ((circumcircle, L)) is the Vietnamese circle of the circumcevian triangles of P and P'; see the preamble just before X₃₉₂₁₀ and centers X₃₉₂₁₉-X₃₉₂₂₈.

  Examples: ((circumcircle, L)), for selected lines L:

  1. L = orthic axis = X₂₃₀X₂₃₂; center of ((circumcircle, L)) is X₆₆₄₄

  2. L = anti-orthic axis = X₄₄X₅₁₃; center of ((circumcircle, L)) is X₃₂₆₁₃

  3. L = Lemoine axis = X₁₈₇X₂₃₇; center of ((circumcircle, L)) is X₁₈₂

  4. L = de Longchamnps axis = X₃₂₅X₅₂₃; center of ((circumcircle, L)) is X₇₅₀₂

  5. L = Nagel line = X₁X₂; center of ((circumcircle, L)) is X₃₉₂₂₅

  6. L = Van Aubel line = X₄X₆; center of ((circumcircle, L)) is X₃₉₂₂₈

  7. L = X₂X₆; center of ((circumcircle, L)) is X₅₉₂₆

  8. L = X₁X₆, center of ((circumcircle, L)) is X₃₉₂₂₇

  9. L = Gergonne line = X₂₄₁X₅₁₄, center of ((circumcircle, L)) is X₃₉₄₇₅

  10. L = Soddy line = X₁X₇, center of ((circumcircle, L)) is X₃₉₄₇₆

  11. L = Fermat line = X₆X₁₃, center of ((circumcircle, L)) is X₃₉₄₇₇

  12. L = X₁X₅, center of ((circumcircle, L)) is X₃₉₄₇₈

  13. L = Sherman line = X₃₂₅₉X₃₃₂₆, center of ((circumcircle, L)) is X₃₉₄₇₉

  14. L = Apollonius line = X₁X₁₈₁, center of ((circumcircle, L)) is X₃₉₂₈₀

  15. L = Napoleon axis = X₆X₁₇, center of ((circumcircle, L)) is X₃₉₂₈₁

  See the preamble just before X₃₉₄₈₆.

• X₃₉₄₈₆–X₃₉₅₅₂ — Centers of circle-inversion circles

  :contributed by Clark Kimberling and Peter Moses, September 2, 2020. Suppose that Γ is the circle with powers (u,v,w) – that is, u = power of A with respect to the circle. and v and w are defined cyclically. Suppose that L is a line, given by p x + q y + r z = 0 (barycentrics). The A-power of the circle ((Γ, L)) is given by (long expression omitted here], and the B- and C- powers are defined cyclically. These powers are useful for finding the center, radius, and equation for the circle ((Γ, L)) [followed by a long list, omitted here].

• X₃₉₆₀₇–X₃₉₆₂₃ — Centers of circle-inversoin circles

  contributed by Clark Kimberling and Peter Moses, September 4, 2020. Suppose that Γ is the circle with powers (u,v,w) – that is, u = power of A with respect to the circle, and v and w are defined cyclically. Suppose that Γ* is the circle with powers (p,q,r). The Γ-inverse of Γ*, denoted by ((Γ, Γ*)), is the circle having powers (f,g,h) given by [long equations and table omittd here].

• X₃₉₆₄₃–X₃₉₆₆₃ — Points associated with Vijay-Paasche-Hutson triangles

  based on notes from Dasari Naga Vijay Krishna, September 10, 2020. In the preambles just before X₃₇₉₉₄, X₃₈₃₈₇, and X₃₉₃₁₁, 18 points, denoted by A_b, A_c, B_c, B_a, C_a, C_b , A'_b, A'_c, B'_c, B'_a, C'_a, C'_b , A″_b, A″_c, B″_c, B″_a, C″_a, C″_b, are defined, along with 31 Vijay-Paasche-Hutson triangles and liests of collinearities and perspectivities. This preamble introduces further such results. The points L_a, P_a, K_a, L'_a, P'_a, K'_a, M_a, N_a, V_a, \(R_{a}\), Q_a defined in the preambles just before X₃₇₉₉₄ and X₃₈₄₈₇ are here relabled as A₁, A₂, A₃, A₄, A₅, A₆, A₇, A₈, A₉, A10, A11, respectively.

  The following sets of points are collinear: {A₁,A₂,A₃}, {A,A16,A30}, [and others]

  Perspectors:of triangles: [list omitted here]

• X₃₉₆₆₇–X₃₉₆₇₈ — Points associated with the 7th Brocard triangle

  based on notes from Dan Reznik and by Peter Moses, September 14, 2020. In the plane of a triangle ABC, let O = X₃, the circumcenter. Let A' be the point, other than O, in which the line AO meets the Brocard circle, and define B' and C' cyclically. The triangle A'B'C', here named the 7th Brocard triangle; the first six Brocard triangles are defined by Bernard Gibert at Brocard triangles and related cubics.

  The 7th Brocard triangle is perspective to the following triangles, with perspector O: ABC, Lucas central, 2nd Hyacinth, and the Lucas antipodal and other Lucas triangles. Also, A'B'C' is perspective to the 2nd Brocard triangle, with perspector X₁₈₄, and to the symmedial triangle, with perspector X₃₉₆₄₃.

  The A vertex of the 7th Brocard triangle is given by

  A' = a^4 + b^4 + c^4 - 2*b^2*c^2 : b^2*(a^2 - b^2 + c^2) : c^2*(a^2 + b^2 - c^2).

  Let KK denote the cubic pK(X₃₉₆₄₄,X₃₉₆₄₅). If P is a point on KK, then the cevian triangle of P is perspective to A'B'C'. The cubic KK passes through the points X_i for i = 3, 6, 230, 5253, 6530, 8770.

  The triangle A'B'C' under the Brocard porism (in which the circumcircle and Brocard inellipse are fixed; see X₃₉) can be viewed at Loci of Centers of Ellipse-Mounted Triangles.) (Dan Reznik, September 14, 2020).

  See also Brocard Porism: locus of 1st, 2nd, 5th, and 7th Brocard Triangles Vertices are Circles.) (Dan Reznik, September 21, 2020).

  Let A″ = circumcircle-inverse of A', and define B″ and C″ cyclically. The triangle A″B″C″, here named the 8th Brocard triangle, is given by

  A″ = 2*a^4 : b^2*(-a^2 + b^2 - c^2) : c^2*(-a^2 - b^2 + c^2)

  B″ = a^2*(a^2 - b^2 - c^2) : 2*b^4 : c^2*(-a^2 - b^2 + c^2)

  C″ = a^2*(a^2 - b^2 - c^2) : b^2*(-a^2 + b^2 - c^2) : 2*c^4

  Note that A″B″C″ are collinear (on the Lemoine axis), so that A″B″C″ is a degenerate triangle.

  Let A* = isogonal conjugate of A″, and define B* and C* cyclically. The triangle A*B*C* is here named the 9th Brocard triangle. Barycentrics are given by

  A* = -((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)) : 2*a^2*(a^2 + b^2 - c^2) : 2*a^2*(a^2 - b^2 + c^2)

  B* = 2*b^2*(a^2 + b^2 - c^2) : -((a^2 + b^2 - c^2)*(-a^2 + b^2 + c^2)) : 2*b^2*(-a^2 + b^2 + c^2)

  C* = 2*c^2*(a^2 - b^2 + c^2) : 2*c^2*(-a^2 + b^2 + c^2) : -((a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2))

  The vertices A*, B*, C* lies on the Steiner circumellipse.

  The 7th Brocard triangle is the dual-of-orthic triangle of the 1st Brocard triangle, and also the anticomplement of the orthic-of-1st-Brocard triangle. (Randy Hutson, September 30, 2020)

  The 9th Brocard triangle is the Steiner-orthic triangle (the Steiner circumellipse counterpart to the circumorthic triangle). (Randy Hutson, September 30, 2020)

• X₃₉₆₉₃–X₃₉₇₄₉ — Cevian Centroid Isogonal Perspectors

  contributed by Vu Thanh Tung, September 18, 2020. Let A₁B₁C₁ be the cevian triangle of a point P =\(p : q : r\) in the plane of a triangle ABC. Let G_a be the centroid of PB₁C₁. Let A' be the isogonal conjugate of G_a with respect to AB₁C₁, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector is given by

  V(P) = a^2 q r (p + q)(p + r)(p + 2q + 2r) : :

  The appearance of (i,j) in the following list means that V(X_i) = X_j: (1,4658), (2,6), (3,49094), (4,631), (5,39667), (6,39668), (7,10389), (8,3361), (9,39669), (10,39670), (31,39671), (32,39672), (75,39673), (76,39674), (83,39675), (141,39676), (560,39677), (561,39678)

  See Cevian Centroid Isogonal Perspector (CevianCentroidIsogonalPerspector.pdf)

• X₃₉₇₆₅–X₃₉₇₇₀ — Points on cubics

  contributed by Clark Kimberling, September 28, 2020.

  Suppose that P =\(p(a,b,c) : p(b,c,a) : p(c,a,b)\) is an polynomial triangle center such that the point (b-c)p(a,b,c) : (c-a)p(b,c,a) : (a-b)p(c,a,b) is on the line at infinity; i.e.,

  (b-c)p(a,b,c) + (c-a)p(b,c,a) + (a-b)p(c,a,b) = 0.

  Let χ(P) denote the cubic given by

  (b+c)p(a,b,c)(y-z)(x+y)(x+z) + (c+a)p(b,c,a)(z-x)(y+z)(y+x) + (a+b)p(c,a,b)(x-y)(z+x)(z+y) = 0.

  This is the cubic pK(X₂),P*), where P* = (b + c)*p - (a + c)*q - (a + b)*r : :

  Let K_aK_bK_c denote the anticomplementary triangle of ABC. It is easy to check that the following 9 points lie on χ(P): A, B, C, K_a, K_b, K_c, X₁, X₂, X₇₅ .

  It is also easy to check that if U is a point on χ(P), then the isotomic conjugate of U also lies on χ(P). This section consists of isotomic conjugates of points on cubics χ(P) for selected points P. In column 1 of the following table, the appearance of u(a,b,c) in a row means that the points indicated in column 2 lie on the cubic

  u(a,b,c)(y-z)(x+y)(x+z) + u(b,c,a)(z-x)(y+z)(y+x) + u(c,a,b)(x-y)(z+x)(z+y) = 0;

  so that the point P is given by u(a,b,c) = (b+c)p(a,b,c). The appearance of {i,j} in column 2 means that {X_i,X_j} are a pair of isotomic conjugate points that lie on χ(P) [table omitted here].

• X₃₉₈₀₁–X₃₉₉₁₃ — Points on cubics

  contributed by Clark Kimberling, October 1, 2020. Let A″B″C″ denote the anticomplementary triangle. The cubic given by

  a(b+c)^2 (b^2+c^2-a^2-bc)(y-z)(x+y)(x+z) + b(c+a)^2 (c^2+a^2-b^2-ca)(z-x)(y+z)(y+x) + c(a+b^2) (a^2+b^2-c^2-ab)(x-y)(z+x)(z+y) = 0

  is a self-isotomic cubic that passes through the following points: A, B, C, A', B', C', X₂, X₇, X₈, and also the six points X₃₉₇₆₅ - X₃₉₇₇₀.

• X₃₉₉₄₂–X₃₉₉₈₄ — More centers related to 7th-, 8th- and 9th- Brocard triangles

  contributed by César Eliud Lozada, October 3, 2020. The 7th-, 8th- and 9th- Brocard triangles are defined in the preamble just before X₃₈₆₄₃. A list of related centers may be seen here (X39801.rtf).

• X₃₉₉₉₄–X₄₀₁₀₆ — Points on cubics

  contributed by Clark Kimberling, October 7, 2020. Suppose that P =\(p(a,b,c) : p(b,c,a) : p(c,a,b)\) is an polynomial triangle center such that the point (b-c)p(a,b,c) : (c-a)p(b,c,a) : (a-b)p(c,a,b) is on the line at infinity; i.e.,

  (b-c)p(a,b,c) + (c-a)p(b,c,a) + (a-b)p(c,a,b) = 0.

  Let ζ(P) denote the cubic given by

  (b+c)p(a,b,c)(cy-bz)(bx+ay)(cx+az) + (c+a)p(b,c,a)(az-cx)(cy+bz)(ay+bx) + (a+b)p(c,a,b)(bx-ay)(az+cx)(bz+cy) = 0.

  Let J_aJ_bJ_c denote the excentral triangle of ABC. It is easy to check that the following 9 points lie on ζ(P): A, B, C, J_a, J_b, J_c, X₁, X₂, X₆ .

  It is also easy to check that if U is a point on ζ(P), then the isotomic conjugate of U also lies on ζ(P). This section consists of isogonal conjugates of points on cubics ζ(P) for selected points P. In column 1 of the following table, the appearance of u(a,b,c) in a row means that the points indicated in column 2 lie on the cubic

  u(a,b,c)(cy-bz)(bx+ay)(cx+az) + u(b,c,a)(az-cx)(cy+bz)(a+bx) + u(c,a,b)(bx-ay)(az+cx)(bz+cy) = 0;

  so that the point P is given by u(a,b,c) = (b+c)p(a,b,c). This is the cubic pK(X₆,P*), where P* = a*((b + c)*p - (a + c)*q - (a + b)*r) : :

  The appearance of {i,j} in column 2 means that {X_i,X_j} are a pair of isogonal conjugate points that lie on ζ(P). [ Table omitted here]

Part 21

• X₄₀₀₀₉–X₄₀₀₄₆ — Points on cubics

  contributed by Clark Kimberling, October 13, 2020; modified January 31, 2021. In column 1 of the following table, the appearance of u(a,b,c) in a row means that the points indicated in column 2 lie on the cubic

  u(a,b,c)(by-cz)(ax+by)(ax+cz) + u(b,c,a)(cz-ax)(by+cz)(by+cz) + u(c,a,b)(ax-by)(cz+ax)(cz+by) = 0.

  The appearance of {i,j} in column 2 means that {X_i,X_j} are a pair of X₅₆₀-isoconjugates that lie on the cubic.

  b+c {2998,6374} (b+c)^2 {18133,40010}, {18140,40013} a^2 (b+c)^2 {17758,18152}, {18137,39735} (b+c)(b+c-a) {18135,40012} (b+c)^2 cos^2 A {18134,40011} (b+c)(2a-b-c) {4358,20568}, {18145,39994}, {39995,40039}, {39996,40040}, {39997,40041} (b+c)(3a+b+c) {18135,40012} (b+c)(-3a+b+c) {18134,40013}, {20934,40026} (b+c)(4a+b+c) {4671,205690} (b+c)(2a+3b+3c) {4359,32018} (b+c)(-a+2b+2c) {18146,40021}, {30829,40029} (b+c)(a^2+b^2+c^2+a(a+b+c)) {870,33931} (b+c)(a+2b+2c)) {19804,40023} (b+c)(bc+ca+ab+a(a+b+c)) {20913,40024} (b+c)(bc+ca+ab-a(a+b+c)) {3948,40017}, {18032,20947} (b+c)(bc+ca+ab-2a(a+b+c)) {31060,40031} (b+c)(2(bc+ca+ab)-a(a+b+c)) {30758,40028}, {30830,40030} (b+c)(abc+a(a^2+b^2+c^2)) {10159,40020}, {33944,40033} (b+c)(2abc+a(a^2+b^2+c^2)) {32000,40032} (b+c)(abc+a(bc+ca+ab)) {10,310}, {4043,40004} (b+c)(abc-a(bc+ca+ab)) {17758,18152} (b+c)(2abc-a(bc+ca+ab)) {20923,40025} (b+c)(b^2c^2+c^2a^2+a^2b^2+a^2bc) {3934,31630} (b+c)(b^2c^2+c^2a^2+a^2b^2-a^2bc) {39,40016} (b+c)(2a+b+c) {274,321}, {35058,40034} (b+c)(abc-a(a^2+b^2+c^2)) {83,8024}, {1031,40035}, {1369,40036}, {20933,40037}, {33938,40038}

• X₄₀₀₅₁–X₄₀₀₇₀ — Dao-perspeconics

  contributed by César Eliud Lozada, October 14, 2020. Let ABC, A'B'C' be two perspective triangles, neither inscribed in the other. Let T₁ be the triangle bounded by the lines BC', CA', AB' and let T₂ be the triangle bounded by the lines BA', CB', AC'. Then the vertices of T₁ and T₂ lie all on a conic (Dao Thanh Oai, October 13, 2020). This conic will be named here the Dao-perspeconic of ABC and A’B’C’.

  The appearance of (T, n) in the following partial list means that the center of the Dao-perspeconic of triangles ABC and T is X_n:

  (ABC-X3 reflections, 3), (anti-Aquila, 40051), (anti-Ara, 40052), (anti-Conway, 15648), (2nd anti-Conway, 15649), (anti-excenters-reflections, 40053),[and others]

  Definitions of all triangles above mentioned can be found in the index of triangles (IndexOfTrianglesReferencedInETC.html).

  < p>

• X₄₀₁₃₉–X₄₀₁₇₃ — Hodpieces

  :This preamble is based on notes received from Radosław Żak (October 29, 2020) and Peter Moses (October 29-30, 2020). In the plane of a triangle ABC, let P be a point, not on a sideline of ABC, and let DEF be the cevian triangle of P. The isogonal conjugate of line EF is a conic. Let A' be the center of of the conic, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in a point here named the hodpiece of P, denoted by H(P). The name hodpiece is taken from James Joyce’s book, Finnegans Wake. The unique point P such that \(H(P) = P\), indexed below as X₄₀₁₃₉, is named the Bloom point after Leopold Bloom, the main character in Joyce’s Ulysses. The point H(X₅) = X₄₀₁₄₀ is the Dedalus point, and the H(X₇) = X₄₀₁₄₁, the Zana point.

  An article (in Polish) about hodpieces by Żak won a gold medal in a competition for high school students organized by the Polish Mathematical Society. For an English translation, see Isogonal conjugate and a few properties of the point X₂₅ (Isogonal%20conjugate%20and%20a%20few%20properties%20of%20the%20point%20X(25).pdf).

  If P =\(p : q : r\) (barycentrics), then \(H(P) = a^{2}/(p (-a^{2}/p + b^{2}/q + c^{2}/r)\): : .

  Let P* = P-Ceva conjugate of X₆. Then \(H(P) = isogonal conjugate of P*-cross conjugate of P\).

  The appearance of (i,j) in the following list means that H(X_i) = X_j: (1,57), (2,25), (3,459), (4,394), (5,40140), (6,2), (7,40141), (9,1422), (10,40142), (13,40156), (14,40157), (15, 40158), (16,40159), [and others]

  Note that H(X₂) = H(X₈₁₁₅) = H(X₈₁₁₆) = X₂₅.

  For Vu Thanh Tung’s generalization to U-hodpieces, see the preamble just before X₄₀₂₁₂.

• X₄₀₁₉₇–X₄₀₂₁₁ — Points associated with Vu parallels conics

  based on notes received from Vu Thanh Tung, October 31, 2020. > In the plane of a triangle ABC, let P =\(p:q:r\) and U =\(u:v:w\) (barycentrics) be points. Let A₁ be the point on BC such that PA₁ is parallel to AU, and define B₁ and C₁ cyclically. Let A₂ be the point on BC such that PA₂ is parallel to AP, and define B₂ and C₂ cyclically. The six points A₁, A₂, B₁, B₂, C₁, C₂ lies on a conic, here named the Vu parallels conic of P and U. See Vu Parallels Conic (VuParallelsConic.pdf).

  Peter Moses (October 31, 2020) found that V(P,U) = [long omitted expression] and T(P,U) = [long omitted expression]

  Let V(P,U) denote the center, and T(P,U) the perspector, of the Vu parallels conic of P and U. The appearance of (i,j,k) in the following list means that V(X_i,X_j) = X_k: (1,2,40197), (1,6,40199), (2,3,40201), (2,4,40203), (2,6,40305), (2,6,40205), (3,4,14767), (3,5,6709), (3,6,40209)

  The appearance of (i,j,k) in the following list means that T(X_i,X_j) = X_k: (1,2,40198), (1,6,40200), (2,3,40202), (2,4,40204), (2,6,40206), (3,4,40207), (3,5,40208), (3,6,40210)

• X₄₀₂₁₂–X₄₀₂₁₈ — U-Hodpieces

  based on notes received from Vu Thanh Tung, November 1, 2020. The definition of hodpiece in the preamble just before X₄₀₁₃₇ generalizes as follows. Let P be a point, not on a sideline of ABC, and let DEF be the cevian triangle of P. Let U =\(u:v:w\) be a point. The P-reciprocal conjugate of U (defined as \(u/p : v/q : w/r\) in the Glossary of ETC), of the line EF is a conic. Let A' be the center of the conic, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in the point

  u / (p*(-u/p + v/q + w/r)) : v / (q*(u/p - v/q + w/r)) : w / (r*(u/p + v/q - w/r)),

  here named the U-hodpiece of P, so that the hodpiece of P is the X₆-hodpiece of P.

• X₄₀₂₃₆–X₄₀₂₉₆ — Tetrahedral projections

  contributed by César Eliud Lozada, November 4, 2020. Let ABC be a triangle on a plane XY. Consider three segments AA', BB', CC' with lengths U, V, W, respectively, and each having one fixed extreme in A, B and C, respectively, and the other extremes free to move outside the plane XY. Suppose that these segments are rotated around their fixed extremes in such a way that their free extremes coincide at a point D, forming, together with the sides of ABC, the edges of a tetrahedron ABCD. Let D* be the orthogonal projection of D on the plane of ABC. The point D* is here named the tetrahedral projection of ABC by (U, V, W) or the tetrahedral projection of ABC to A’B’C’. Then

  D* = \(a^{2}\) (SA - \(U^{2})\) + SB \(W^{2}\) + SC \(V^{2}\) : \(b^{2}\) (SB - \(V^{2})\) + SC \(U^{2}\) + SA \(W^{2}\) : \(c^{2}\) (SC - \(W^{2})\) + SA \(V^{2}\) + SB \(U^{2}\) (1)

  Z(D), the Z-coordinate of D , i.e., the height of the point D measured from D* and orthogonally to the plane of ABC, is given by:

  Z(D) = ±\(\sqrt{∑ [2 (a^{2} U^{2} + V^{2} W^{2}) S_{A} - a^{2} U^{4}] - (a b c)^{2}}\)/(2 S) (2)

  Equation (2) shows that D is real or imaginary according to the sign of the quantity under the square root. If this quantity is zero then D and D* coincide on the plane of ABC. Moreover, the ± sign indicates that there are two possible points D and D', each in different sides with respect to the plane of ABC.

  Equation (1) shows that if U, V, W are real numbers then D* is always real and also that, if U, V, W are cyclic values, i.e., if there exists a degree-1 function ƒ(a,b,c) such that U=ƒ(a,b,c), V=ƒ(b,c,a) and W=ƒ(c,a,b), then D* is a triangle center.

  The appearance of (ƒ(a,b,c), n) in the following list means that the tetrahedral projection of ABC by (U=ƒ(a,b,c), V=ƒ(b,c,a), W=ƒ(c,a,b)) is X_n:

  (a, 20), (b+c, 962), (-a+b+c, 8148), (\(\sqrt{b^{2}+c^{2}}\), 4), (\(\sqrt{-a^{2}+b^{2}+c^{2}}\), 382) The appearance of (i, j) in the following list means that the tetrahedral projection of ABC to the cevian triangle of X_i is X_j: (2, 546), (4, 389), (7, 65), (39158, 3), (39159, 3) The appearance of (i, j) in the following list means that the tetrahedral projection of ABC to the anticevian triangle of X_i is X_j: (1, 1490), (2, 3146), (6, 40285) The appearance of (i, j) in the following list means that the tetrahedral projection of ABC to the circumcevian triangle of X_i is X_j: (1, 40257), (3, 3), (4, 6241), (186, 74), (8697, 8697) The appearance of (i, j) in the following list means that the tetrahedral projection of ABC to the circumanticevian triangle of X_i is X_j: (1, 40256), (2, 12111), (6, 3) The appearance of (T, n) in the following list means that the tetrahedral projection of ABC to the triangle T is X_n:

  (ABC-X3 reflections, 3), (anti-Aquila, 13624), (anti-Ara, 1885), (anti-Artzt, 5077), [and others]

  Definitions of all triangles above mentioned can be found in the index of triangles (IndexOfTrianglesReferencedInETC.html).

  Preamble edited on June 28, 2022.

• X₄₀₂₉₇–X₄₀₃₀₅ — Points associated with the power curve

  contributed by Suren, November 4, 2020. In the plane of a triangle ABC, the locus of a point \(a^{t} : b^{t} : c^{t}\) (barycentrics [or trilinears]) as t varies through the real numbers is the power curve, PC(ABC), of ABC. (The term is introduced in Clark Kimberling, “Major Centers of Triangles,” American Math. Monthly 104 (1997), 431-438.) Note that PC(ABC) passes through X_i for i = 1,2,6,31,75,76, and that eliminating t shows that PC(ABC) is given by the equations

  (log x)/(log a) = (log y)/(log b) = (log z)/(log c).

  (Here, “log” signifies the natural logarithm, but equivalent equations result under change of base for “log”.) Centers X₄₀₂₉₇-X₄₀₃₀₅ involve the line tangent to PC at X₁, X₂, and X₆.

  In general, the line tangent to the power curve at a point \(a^{t} : b^{t} : c^{t}\) has the direction (i.e., a point on the infinity line) given by

  (a*c)^t log(a/c) + (a*b)^t log(a/b) : : ,

  and the trilinear pole of that point is the point \(a^{t}\) log(c/b) : \(b^{t}\) log(a/c) : \(c^{t}\) log(b/a).

• X₄₀₃₀₆–X₄₀₃₁₅ — Points on Vu orthogonal conics

  based on notes contributed by Vu Thanh Tung, November 5, 2020. In the plane of a triangle ABC, let P and U be points. Let L be the line through P perpendicular to line AU, and let A₁ = L∩BC. Define B₁ and C₁ cyclically. Let L' be the line through U perpendicular to line AP. and A₂ = L'∩BC. Define B₂ and C₂ cyclically. The six points A₁, B₁, C₁, A₂, B₂, C₂ lie on a conic, here named the Vu orthogonal conic of P and U, denoted by VOC(P,U).

  Let V(P,U) denote the center, and T(P,U) the perspector, of VOC(P,U). Note that VOC(U,P) = VOC(P,U), V(U,P) = V(P,U), and T(U,P) = T(P,U).

  See Vu Orthogonal Conic (VuOrthogonalConic.pdf).

• X₄₀₃₂₈–X₄₀₃₃₆ — Osiris points

  contributed by César Eliud Lozada, November 9, 2020. Let ABC be a triangle, P a point and Q the isotomic conjugate of P. Denote by A' the centroid of the quadrangle BCPQ and define B' and C'. cyclically. Then AA', BB', CC' concur at a point O(P)=O(Q), here named the Osiris point of P.

  For P=\(x:y:z\) (barycentrics), O(P) = (y^2+5*y*z+z^2)*x+2*(y+z)*(x^2+y*z) : :

  The appearance of (i, j) in the following list means that the Osiris point of X_i is X_j:

  (1, 40328), (2, 2), (3, 40329), (4, 40330), (5, 40331), (6, 40332), (7, 40333), (8, 40333), (13, 40334), (14, 40335), (69, 40330), (75, 40328), (76, 40332), [and others]

  If P or Q lie on the cubic K953, then its Osiris point lies on the Euler line of ABC.

  The mapping O takes certain cubics onto lines: O(K296) = X₁X₂, O(K185) = X₂X₆, O(K953) = X₂X₃. (Peter Moses, November 10, 2020)

• X₄₀₃₄₅–X₄₀₃₄₆ — Points on the tangential power curve

  contributed by Clark Kimberling, November 11, 2020. Let PC(ABC) by the power curve; i.e., the locus of the point \(a^{t} : b^{t} : c^{t}\) (barycentrics) as t varies through the real numbers. If P =\(a^{t} : b^{t} : c^{t}\) for fixed t, then the isotomic conjugate of P is the point P' =\(a^{-t} : b^{-t} : c^{-t}\). Equations for the lines tangent to PC(ABC) at P and P' are found using Suren’s points on the line at infinity; see the preamble just before X₄₀₂₉₆. If P is not the centroid of ABC, then the tangent lines are distinct, and they meet in the point

  P″ = \(a^{t}(b^{2t} - c^{2t})\)/(log b - log c) : \(b^{t}(c^{2t} - a^{2t})\)/(log c - log a) : \(c^{t}(a^{2t} - b^{2t})\)/(log a - log b)

  The locus of P″ as t varies through the positive real numbers is here named the (barycentric) tangential power curve.

  The corresponding normal lines at P and P' meet in a point whose locus is the normal power curve.

  The trilinear tangential and normal power curves are defined in the same manner using trilinear coordindates throughout, using isogonal conjugates instead of isotomic.

• X₄₀₃₅₉–X₄₀₃₇₅ — Ceva-conjugates associated with the power curve

  contributed by Clark Kimberling and Peter Moses, November 17, 2020. Let P(t) =\(a^{t} : b^{t} : c^{t}\), on the power curve, as in the preambles just before X₄₀₂₉₇ and X₄₀₃₄₅. The P(t)-Ceva conjugate of P(u), denoted by P(t)©P(u) is given by

  \(a^{u}(-a^{u-t} + b^{u-t} + c^{u-t})\) : \(b^{u}(a^{u-t} - b^{u-t} + c^{u-t})\) : \(c^{u}(a^{u-t} + b^{u-t} - c^{u-t})\),

  P(t)©P(u) is the perspector of the cevian triangle of P(t) and the anticevian triangle of P(u).

  The appearance of (i,j,k) in the following list means that P(t)©P(u) = X_k:

  (-9,-1,33807), (-8,-8,40359), (-8,-6,40360), (-8,-2,40361), (-8,0,33797), (-7,-1,33806), (-7,1,33791), (-6,-6,40362), (-6,-4,40050), [and others]

  For fixed t = \(t_{0}\) and variable u, the locus of \(P(t_{0})\)©P(u) is here named the \(P(t_{0})\)©P(u)-Ceva power curve. For fixed variable t and fixed u = \(u_{0}\), the locus of P(t)©\(P(u_{0})\) is here named the P(t)©\(P(u_{0})\)-Ceva power curve.

• X₄₀₃₇₆–X₄₀₃₈₃ — Complements and anticomplements associated with the power curve

  contributed by Clark Kimberling and Peter Moses, November 19, 2020. Suppose that P(t) =\(a^{t} : b^{t} : c^{t}\) (barycentrics) is a point on the power curve. The complement of P(t) is the point \(b^{t}\) + \(c^{t}\) : \(c^{t}\) + \(a^{t}\) : \(a^{t}\) + \(b^{t}\). The anticomplement of P(t) is the point -\(a^{t}\) + \(b^{t}\) + \(c^{t}\) : \(a^{t}\) - \(b^{t}\) + \(c^{t}\) : \(a^{t}\) + \(b^{t}\) - \(c^{t}\).

  The appearance of (i,j) in the following list means that the (complement of X_i) = X_j:

  (-8,40376), (-6,40377), (-4,8265), (-3,16584), (-2,39), (-1,37), (-1/2,40378), (0,2), (1/2,20527), (1,10), (3/2,20334), (2,141), (5/2,20543), (3,2887), (4,626), (5,21235), (6,40379), (8,40380)

  The appearance of (i,j) in the following list means that the (anticomplement of X_i) = X_j:

  (-8,40381), (-6,40382), (-4,8264), (-3,17486), (-2,194), (-1,192), (-1/2,40383), (0,2), (1/2,20534), (1,8), (3/2,20346), (2,69), (5/2,20555), (3,6327), (4,315), (5,21275), (6,33796), (8,33797)

• X₄₀₄₆₉–X₄₀₅₂₉ — Centers and perspectors of cevapoint conics

  contributed by Clark Kimberling and Peter Moses, November 30, 2020. In the plane of a triangle ABC, let L be the line u x + v y + w z = 0, and let U be the point \(u : v : w\), this being the isotomic conjugate of the trilinear pole of L. Let P =\(p : q : r\) be a point. The (U,P)-cevapoint conic, introduced here as the locus of X such that the cevapoint of P and X is on the line LU is given by

  u (q x + p y)(r x + p z) + v (r y + q z)(p y + q x) + w (p z + r x)(q z + r y) = 0.

  The center of the conic is the point

  p*(p^2*(p - q - r) u^2 + q^2 (p - q + r) v^2 + r^2 (p + q - r) w^2 - 2 p q r v w + 2 p r (p - r) w u + 2 p q (p - q) u v) : : ,

  and the perspector, by

  p/(-p u + q r + r w) : q/(p u - q v + r w) : r/(p u + q v - r w).

  For every point U, the (U,P)-cevapoint conic passes through the vertices of the anticevian triangle of P.

  The appearance of (i,j,k) in the following list means that the center of the (X_i,X_j)-cevapoint conic is X_k:

  (1,1,15487), (1,2,7), (1,37,10), (1,514,693), (1,661,523), (1,1577,850), (1,3239,4397), (2,2,2), (6,6,14713), (2,1,40), (2,6,159), (2,37,22271), [and others]

  The appearance of {i, {j(1),j(2),…}} in the following list means that the (X₁,X_i)-cevapoint conic passes through the points X(j1), X(j2),… :

  {1, {2640,5540,16550,16559,16560,16561,16562,16563}}

  {2, {149,4440,20355,20533,21220,21221,30578,37781}}

  {6, {2932,20871,20999,21004,23402,23860}}

  {10, {21090,21100,22029,22031,22035}} [and others]

  The appearance of {i, {j(1),j(2),…}} in the following list means that the (X₂,X_i)-cevapoint conic passes through the points X(j1), X(j2),… :

  {1, {1054,1282,1768,2100,2101,2448,2449,2948,3464,5539,5540,5541,9860,9904,12408,13174,13221,13513,20114,20375,21381,34196,34464,39156}}

  {2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}

  {6,{2930,7669,10117,15588,16686,20468,20998,20999,23858}}

  {37, {20694,21889,21893,22313,22323}}[and others]

  The appearance of {i, {j(1),j(2),…}} in the following list means that the (X₇₅,X_i)-cevapoint conic passes through the points X(j1), X(j2),…:

  {1, {1054,2629,2636,2640,9324,9355,9359,39335,39336,39337,39338,39339,39340,39341,39342,39343,39344}}

  {6, {3196,9259,9509,16686,20672,21004,21783}}

  {512, {661,798,3709,14090}}

  [and others]

  The appearance of {i, {j(1),j(2),…}} in the following list means that the (X₆,X_i)-cevapoint conic passes through the points X(j1), X(j2),…:

  {1, {16560,16565,20601,21381,21382,39335}}

  {2, {146,147,148,149,150,151,152,153,3448,11671,12384,13219,13510,14360,14731,14732,14807,14808,20344,21290,33650,34186,34188,34193,34547,34548,34549,34550}}

  {6, {2936,7669,16873,23402,39857}}

  [and others]

  The appearance of {i, {j(1),j(2),…}} in the following list means that the (X₁₀,X_i)-cevapoint conic passes through the points X(j1), X(j2),…:

  {1, {5540,9359,16554,24578}}

  {2, {4440,17154,21224,30579,33888}}

  {6, {8301,9259,20999,23392,23404}}

  {513, {514,649,650,4083,6589,14079}}

  {514, {513,514,905,14078,14079,21172,21191,21194}}

  {522, {650,4521,14837,20317}}

  {649, {513,649,1459,14079,14088}}

  The appearance of {i, {j(1),j(2),…}} in the following list means that the (X₅₁₄,X_i)-cevapoint conic passes through the points X(j1), X(j2),…:

  {1, {1,9,40,188,191,366,1045,1050,1490,2136,2949,2950,2951,3174,3307,3308,3646,5506,5528,5541,6326,12658,12660,13144,13146,16009,16550,16558,18598,24578,25427,32632,38004,39131}}

  {2, {2,144,192,366,1654,3151,4182,17487,17488,20533,24313,24314,27484,31308,33888,37881}}

  {3, {6,3157,7078,22133}}

  {6, {3,55,197,199,8301,11505,11506,12335,18755,20871,20996,23858,23859,36943}}

  {8, {8,188,3161,6731,8834,19582,30412,30413,39800}}

  {9, {1,200,3158,7070}}

  {10, {10,37,72,3159,8804,20722,21080,21083,22271,22299,22306,22307,39131}}

  [and others]

  The appearance of {i, {j(1),j(2),…}} in the following list means that the (X_i,X_i)-cevapoint conic passes through the points X(j1), X(j2),… :

  {1, {2640,5540,16550,16559,16560,16561,16562,16563}}

  {2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}

  {6, {2936,7669,16873,23402,39857}}

  {514, {1086,4904,14078,17761,24185}}

  {523, {115,5461,6128,7668,14086,39022,39023}}

  {525, {127,2454,2455,15526}}

  Let X*(i) denote the isotomic conjugate of X_i. The appearance of {i, {j(1),j(2),…}} in the following list means that the (X*(i),X_i)-cevapoint conic passes through the points X(j1), X(j2),…:

  {1, {1054,2629,2636,2640,9324,9355,9359,39335,39336,39337,39338,39339,39340,39341,39342,39343,39344}}

  {2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}

  {3, {20795,22143,22148,22158,23081,23180}}

  {6, {1979,9259,9412,9431,20998,21781}}

  {37, {21885,21888,21893,21899}}

  {514, {14078,21200,21204,21211}}

  Let X^2(i) denote the barycentric square of X_i. The appearance of {i, {j(1),j(2),…}} in the following list means that the (X^2(i),X_i)-cevapoint conic passes through the points X(j1), X(j2),…:

  {1, {16560,16565,20601,21381,21382,39335}}

  {2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}

  {30, {2,402,23583,24975}}

  {512, {2,3589,4698,6375,6387,6677,6685,6719,14090,15895,15896,34236}}

  {513, {2,1125,6692,6703,6714,14079,16604,28600,36812}}

  [and others]

• X₄₀₅₃₀–X₄₀₅₆₄ — Centers and perspectors of 1st Ceva conics

  contributed by Clark Kimberling and Peter Moses, November 30, 2020. In the plane of a triangle ABC, let L be the line u x + v y + w z = 0, and let U be the point \(u : v : w\), this being the isotomic conjugate of the trilinear pole of L. Let P =\(p : q : r\) be a point. The 1st (U,P)-Ceva conic, introduced here as the locus of X such that the P-Ceva conjugate of X is on the line L, is given by

  u p (-p/x + q/y + r/z) + v q (p/x - q/y + r/z) + w r (p/x + q/y - r/z) = 0.

  The center of the conic is the point

  p (-p q^2 u v - 3 p q r u v + p q^2 v^2 - q^2 r v^2 - 3 p q r u w - p r^2 u w - 2 p q r v w - 3 q^2 r v w - 3 q r^2 v w + p r^2 w^2 - q r^2 w^2) : :

  If U = X₂, then the center of the 1st (U,P)-Ceva conic is the complement of the complement of P, which is also the centroid of {A, B, C,P}, and also the center of the bicevian conic of X₂ and P. (Randy Hutson, December 18, 2020)

  The appearance of (i,j,k) in the following list means that the center of the 1st (X_i,X_j)-Ceva conic is X_k:

  (2,1,1125), (2,2,2), (2,3,140), (2,4,5), (2,5,3628), (2,6,3589), (2,7,142), (2,8,10), (2,10,3634), (2,25,6677), [and others]

  The appearance of {i, {j(1),j(2),…}} in the following list means that the 1st (X₂,X_i)-Ceva conic passes through the points X(j1), X(j2),… :

  {1, {11,214,244,1015,8054,8299,10494,14714,17417,17419,17421,17761,17793,34586,34587,34588,34589,34590,34591,34592,34593,38978,38979,38980,38981,38982,38983,38984,38985,38986,39046}

  [and others]
  The appearance of {i, {j(1),j(2),…}

  in the following list means that the 1st (X₁,X_i)-Ceva conic passes through the points X(j1), X(j2),… :
  {2, {244,1086,2968,4858,5515,6377,16586,17755,38995,39040}

  {75, {244,1099,1109,1111,4712,4736,4738,10504,17879,23996,24010,24014,24023,24026,24028,24031,24034,24038}

  {76, {1086,1111,1227,3123,21208,34387}

  [and others]

  The appearance of {i, {j(1),j(2),…} in the following list means that the 1st (X₇₅,X_i)-Ceva conic passes through the points X(j1), X(j2),… :

  {1, {244,678,2310,2632,2638,2643,3248,4094,4117,10501,23063,24012}

  [and others]
  The appearance of {i, {j(1),j(2),…}

  in the following list means that the 1st (X₁₀,X_i)-Ceva conic passes through the points X(j1), X(j2),… :
  {2, {1086,3120,6627,6651,8054,16726,38960}

  {99, {1,21,86,1125,2309,8053,18650,28627}

  [and others]
  The appearance of {i, {j(1),j(2),…}

  in the following list means that the 1st (X₇₅,X_i)-Ceva conic passes through the points X(j1), X(j2),… :

  {1, {1,37,192,2292,2667,3057,3159,4065,4319,5497,8393,8394,17460,17461,17464,17475,18674,19582,23757,34587,39916} [and others]

  The appearance of {i, {j(1),j(2),…} in the following list means that the 1st (X₁,X_i)-Ceva conic passes through the points X(j1), X(j2),… :

  {2, {115,1015,1084,1086,1146,2454,2455,2482,3163,4370,5997,6184,11672,13466,15166,15167,15449,15525,15526,15527,17416,17429,18334,20532,23967,23972,23976,23980,23986,23992,35066,35067,35068,35069,35070,35071,35072,35073,35074,35075,35076,35077,35078,35079,35080,35081,35082,35083,35084,35085,35086,35087,35088,35089,35090,35091,35092,35093,35094,35095,35110,35111,35112,35113,35114,35115,35116,35117,35118,35119,35120,35121,35122,35123,35124,35125,35126,35127,35128,35129,35130,35131,35132,35133,35134,35135,35508,35509,39008,39009,39010,39011,39012,39013,39014,39015,39016,39017,39018,39019,39020,39021,39022,39023,39206,39207,39208,39209}

  {7, {3022,3271,4904,26932}

  {99, {148,2482,7669,12076,14443}

  {100, {16560,17060,22308,23402}

  {190, {4370,4440,5540,14442,17464,22035}

  {513, {9263,13466,14441,22323}
  Let X*(i) denote the isotomic conjugate of X_i. The appearance of {i, {j(1),j(2),…}

  in the following list means that the 1st (X*(1),X_i)-Ceva conic passes through the points X(j1), X(j2),… :

  {1, {244,678,2310,2632,2638,2643,3248,4094,4117,10501,23063,24012} [and others]

• X₄₀₅₇₈–X₄₀₆₂₉ — Centers of 2nd Ceva conics

  contributed by Clark Kimberling and Peter Moses, December 6, 2020. In the plane of a triangle ABC, let L be the line u x + v y + w z = 0, and let U be the point \(u : v : w\), this being the trilinear pole of L. Let P =\(p : q : r\) be a point. The 2nd (U,P)-Ceva conic is introduced here as the locus of X such that the X-Ceva conjugate of P is on the line L. This conic circumscribes ABC and is given by

  p(-u p + v q + w r)/x + q(u p - v q + w r)/y + r(u p + v q - w r)/z = 0.

  The center of the conic is the point

  p*((p+q+r) p^2 u^2 + (p+q-r) q^2 v^2 + (p-q+r) r^2 w^2 + 2 p q r v w - 2 p r (p+r) w u - 2 p q (p+q) u v) : :

  If U = X₂, then the center of the 2nd (U,P)-Ceva conic is the X₂-Ceva conjugate of P, and the perspector of the 2nd (U,P)-Ceva conic is P. (Randy Hutson, December 18, 2020)

• X₄₀₆₃₂–X₄₀₆₆₁ — Points associated with bicevian triangles

  contributed by Clark Kimberling and Peter Moses, December 8, 2020. Let P =\(p : q : r\) and U =\(u : v : w\) be points in the plane of a triangle ABC. Let A'B'C' be the cevian triangle of P and A″B″C″ the cevian triangle. Let A* be the midpoint of A' and A″, and define B* and C* cyclically. The triangle A*B*C* is here named the (P,U)-bicevian triangle:

  A* = 0 : 2 q v + q w + r v : 2 r w + q w + r v

  B* = 2 p u + r u + p \(w : 0 : 2\) r w + r u + p w

  C* = 2 p u + p v + q u : 2 q v + p v + q u : 0

  For example, let DEF be the (X₂,X₄)-bicevian triangle, so that D = 0 : 2 a^2 + b^2 - c^2 : a^2 - b^2 + c^2. The vertices D, E, F lie on the cubics K054 and K124 and on the Moses-Steiner ellipse (see X₆₀₇₀.

  These points lie on the Euler line of DEF:

  X₅₉₄₃ = X₂-of-DEF

  X₈₂₅₄ = X₃-of-DEF

  X₆₁₅₃ = X₄-of-DEF

  X₁₃₃₆₅ = X₅-of-DEF

  X₄₀₆₃₂ = X₂₀-of-DEF

  The following triangles are perspective to DEF, all with perspector X₅: 3rd and 4th Euler triangles, submedial, infinite altitude, Ehrmann mid-triangle, Gemini 110, 1st and 2nd half-diamonds equilateral triangles, and 1st and 2nd half-diamonds triangles (X₃₃₃₃₈.

  The circumcircle (M), of DEF, passes through X_i for i = 125, 137, 11702, 14071, 30480 and has squared radius

  (2*(a^2 + b^2 - c^2)^2 - a^2*b^2*(-2 + J^2))*(2*(a^2 - b^2 + c^2)^2 - a^2*c^2*(-2 + J^2))*(2*(-a^2 + b^2 + c^2)^2 - b^2*c^2*(-2 + J^2))/(64*a^2*b^2*c^2*(-2 + J)^2*(2 + J)^2*S^2)

  Note that (M) meets the nine-point circle in the points X₁₂₅ and X₁₃₇.

  DEF is the reflection triangle of the medial-of-medial triangle (which is also Gemini triangle 110, the X₂-midcevian triangle, and the excentral triangle of the submedial triangle, if ABC is acute), and DEF is homothetic to the reflection triangle at X₂. (Randy Hutson, December 18, 2020)

  Let D' be the point, other than D, where (M) meets the line BC, and define E' and F' cyclically. Then

  Let D' = 0 : (a^2 + 2*b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2) : (a^2 - b^2 + 2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4).

  The triangle D'E'F', here named the Maia triangle, is perspective to the following triangles, with perspectors as shown:

  1st orthosymmedial (see X₆₇₉₂; perspector X₄₀₆₃₃

  infinite altitude; perspector X₅₄

  orthic axes triangle (see X₂₅₀₁₀); perspector X₂₇₅

  Yiu tangents triangle (see X₇₄₉₅; perspector X₄₀₆₃₄

  The Maia triangle is homothetic to the polar triangle of the nine-point circle at X₈₉₀₁. (Randy Hutson, December 18, 2020)

• X₄₀₆₇₀–X₄₀₆₇₃ — Points associated with the pedal triangle of the centroid

  contributed by Clark Kimberling and Peter Moses, December 9, 2020. Let T denote the pedal triangle of X₂; T is perspective to these triangles:

  orthocentroidal, with perspector X₁₉₉₂

  1st Ehrmann, with perspector X₁₉₉₅

  Artzt, with perspector X₂

  infinite altitude, with persector X₂

  anti-Artzt, with perspector X₂

  Gemini 105 triangle, with perspector X₁₄₅

  Gemini 107 triangle, with perspector X₁₉₉₂
  X₂-of-T = X₃₇₃

  X₃-of-T = X₅₉₇

  X₄-of-T = X₂₉₉₅₉

  X₅-of-T = X₄₀₆₇₀

  X₆-of-T = X₃₃₆₃

  X₁₅-of-T = X₄₀₆₇₁

  X₁₆-of-T = X₄₀₆₇₂

  X₂₀-of-T = X₄₀₆₇₃

  X₃₀-of-T = X₃₈₅₄

• X₄₀₆₇₄–X₄₀₆₉₆ — Points associated with midcevian triangles

  contributed by Clark Kimberling and Peter Moses, December 9, 2020. Let U =\(u : v : w\) be a point in the plane of a triangle ABC. Let A'B'C' be the cevian triangle of U, and let A″ be the midpoint of the segment AA'. Define B″ and C″ cyclically, so that

  A″ = v +\(w : v : w\)

  B″ = u : w + u : w

  C″ =\(u : v : u\) + v.

  The triangle A″B″C″ is here named the U-midcevian triangle. Examples include

  X₁-midcevian triangle = Gemini triangle 15

  X₂-midcevian triangle = Gemini triangle 110

  X₄-midcevian triangle = half-altitude triangle

  X₇-midcevian triangle = 1st Zaniah triangle

  X₈-midcevian triangle = 2nd Zaniah triangle

  X₆₉-midcevian triangle = orthic-of-medial triangle = anti-6th-mixtilinear = anticomplement of submedial triangle; see X₁₁₃₆₃

  X₇₅-midcevian triangle = Gemini triangle 16 = complement of incentral triangle = n(Incentral)*n(Medial) (ETC preamble before X₃₇₃₉)

  X₅₂₃-midcevian triangle = anticevian triangle of X₅₂₃ = Schroeter triangle (ETC X₈₂₈₆ = diagonal triangle of Feuerbach quadrangle of ABC (ETC X₁₀₂₇₆

  In general, the U-midcevian triangle is perspective to the following triangles:

  ABC, with perspector U

  medial triangle, with perspector u v + u w : :

  Wasat triangle (see X₂₁₆₁₆), with perspector a u (b + c) - (b v - c w)(b - c) : :

  Gemini 7 triangle, with perspector a u (a - b - c) - (b - c)((a - b + c) v + (a + b - c) w) : :

  Let T(U) denote the midcevian triangle of U, and let C(U) denote the cevian triangle of U.

  The locus of a point X such that T(U) is perspective to C(X) is the cubic pK(X₂,U*), where U* = isotomic conjugate of U.

  The locus of X such that T(U) is perspective to the anticevian triangle of X is the cubic pK(u*(v + w) : : , -u + v + w : :). For example, if U = X₃, then the cubic is K044.

  The locus of X such that T(X₁) is perspective to C(X) is the cubic K034.

  The locus of X such that T(X₃) is perspective to C(X) is the cubic K045.

  The locus of X such that T(X₄) is perspective to C(X) is the cubic K007.

  The locus of X such that T(X₆) is perspective to C(X) is the cubic K141.

  The locus of X such that T(X₇) is perspective to C(X) is the cubic K200.

  The locus of X such that T(X₈) is perspective to C(X) is the cubic K1078.

  The locus of X such that T(X13)) is perspective to C(X) is the cubic K264a.

  The locus of X such that T(X₁₄) is perspective to C(X) is the cubic K264b.

• X₄₀₇₁₈–X₄₀₇₂₅ — Points associated with CCC cubics

  contributed by Clark Kimberling and Peter Moses, December 16, 2020. Let P =\(p : q : r\) and U =\(u : v : w\) be points in the plane of a triangle ABC, and let

  A'B'C' = cevian triangle of P, D'E'F' = cevian triangle of U

  A″B″C″ = anticevian triangle of P, D″E″F″ = anticevian triangle of U

  A* = A'D″ ∩ A″E', and define B* and C* cyclically, so that
  A* = - p u : q u + p v : r u + p w

  B* = p v + q u : - q v : r v + q w

  C* = p w + r u : q w + r v : - r w

  The triangle A*B*C* is here named the (P,U)-cevian-cross triangle (not to be confused with the cross-cevian triangle in TCCT, p. 201)..

  The locus of a point X =\(x : y : z\) such that the (P,U)-cevian-cross triangle is perspective to the cevian triangle of X is the (P,U)-CCC cubic, given by

  (r u + p w)(q^2 u^2 + p q u v + p^2 v^2) y z^2 - (q u + p v) (r^2 u^2 + p r u w + p^2 w^2) y^2 z + (cyclic) = 0

  The (P,U)-CCC cubic is the cubic pK(P*,U*), where

  P* = q^2 u^2 + p q u v + p^2 v^2)(r^2 u^2 + p r u w + p^2w^2) : :

  U* = (q^2 u^2 + p q u v + p^2*v^2)(r^2 u^2 + p r u w + p^2 w^2)(r v + q w) : :
  Examples:

  (X₁₅, X₁₆)-CCC cubic = pK(X₆, X₃₀) = K001

  (X₂, X₆)-CCC cubic = pK(X₃₄₀₇, X₁₄₆₁₇) = K421

  The locus of a point X =\(x : y : z\) such that the (P,U)-cevian-cross triangle is perspective to the anticevian triangle of X is the (P,U)-CCA cubic, given by

  2 p u (r^2 u v + 2 q r u w + 2 p r v w + p q w^2) y^2 z - 2 p u (2 q r u v + p r v^2 + q^2 u w + 2 p q v w) y z^2 + (cyclic) = 0.

  The (P,U)-CCA cubic is the cubic pK(P*,U*), where

  P* = p u : : , and U* = q r u^2 + p^2 v w + 2 p u (r v + q w)) : :

  Examples:

  ((X₂, X₄)_CCA cubic = pK(X₄, X₄₅₈) = K677

  ((X₂, X₆)_CCA cubic = pK(X₆, X₃₃₂₉) = K423

  ((X₂, X₁₃)_CCA cubic = pK(X₁₃, X₈₈₃₈) = K420b

  ((X₂, X₁₄)_CCA cubic = pK(X₁₃, X₈₈₃₆) = K420a

  ((X₂, X₃₀)_CCA cubic = pK(X₃₀, X₂) = K472

  ((X₆, X₉₈)_CCA cubic = pK(X₁₉₇₆, X₉₈) = K380

  ((X₁₃, X₁₄)_CCA cubic = pK(X₁₉₈₉, X₂₆₅) = K060

  ((X₁₅, X₁₆)_CCA cubic = pK(X₅₀, X₃) = K073

• X₄₀₈₄₀–X₄₀₈₄₂ — Points associated with (X₁,U)-cevian-cross triangles

  contributed by Clark Kimberling and Peter Moses, December 31, 2020. Let U =\(u : v : w\) be a point in the plane of a triangle ABC. The (X₂,U)-cevian-cross triangle A'B'C' is introduced in the preamble just before X₄₀₇₁₈. The vertices are given by.

  A″ = u : u + v : u + w

  B″ = v +\(u : - v : v\) + w

  C″ = w + u : w + v : - w.

  Triangles perspective to A'B'C' and the perspectors include the following:

  ABC, (u+v)*(u+w) : :

  medial, u*(u - v - w) : :

  anticomplementary, u*v + u*w - v*w : :

  [and others]

• X₄₀₈₄₃–X₄₀₈₅₀ — Points associated with paratriangles

  contributed by Clark Kimberling and Peter Moses, December 31, 2020. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. The point P' =\(p : r : q\) is here named the parapoint of P. To contruct P', let L be the line through P parallel to BC. Let Q = L∩AG , where G denotes the centroid of ABC. Then P' is the reflection of P in Q.

  The paratriangle of P is the triangle A'B'C' with vertices given by

  A' =\(p : r : q\)

  B' =\(r : q : p\)

  C' =\(q : p : r\)

  If P is a triangle center other than X₁, then A'B'C' is a noncentral triangle, but its perspector with other triangles can be a triangle center.

  (More generally, the paratriangle of a triangle UVW is here defined as the triangle U'V'W'.)

  For fixed P with paratriangle A'B'C', the locus of X such that the cevian triangle of X is perspective to A'B'C' is the cevian-associated cubic given by

  q (p q - r^2) y z^2 - r (p r - q^2) y^2 z + (cyclic) = 0,

  which is the cubic pK(P₁,P₂), where P₁ = (p r - q^2)(p q - r^2) and P₂ = P*P₁.

  The appearance of (i,K___) in the following list means that if P = X_i, then the K___ is the cevian-associated cubic;

  (1, K768)

  (4, K776)

  (6, K322)

  (13, K859b)

  (14, K859a)

  (30, K860)

  [and others]

  The following list shows, as an example, that if P = X₇, then the points listed lie on the cevian-associated cubic:

  {7,{2,8,9,239,673,2319,2481,4373,5853,9312,9436,14942,33676}}

  {8,{1,2,7,516,673,2481,3729,3912,6185,10025,10405,14942,39914}}

  {10,{2,86,350,1125,6542,6625,6650,9505,11599,17770,20536,28604,39921}}

  [and others]

  The next list shows the points P₁ and P*P₁ for each cubic in the list just above. For example, if P = X₇, then the cevian-associated cubic is pK(X₁₄₉₄₂, X₆₇₃₁₂).

  {7,{14942,673}}

  {8,{673,14942}}

  {10,{6650,11599}}

  [and others]

• X₄₀₈₅₃–X₄₀₈₉₃ — Points associated with paratriangles

  contributed by Clark Kimberling and Peter Moses, January 1, 2021. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. As in the preamble just before X₄₀₈₄₃, the paratriangle of P is the triangle A'B'C' with vertices given by

  A' =\(p : r : q\)

  B' =\(r : q : p\)

  C' =\(q : p : r\)

  For fixed P with paratriangle A'B'C', the locus of X such that the anticevian triangle of X is perspective to A'B'C' is the anticevian-associated cubic given by

  p (p r - q^2) y z^2 - p (p q - r^2) y^2 z + (cyclic) = 0,

  which is the cubic pK(P,P'), where P' = p^2 - q r : q^2 - r p : r^2 - p q is the X₂-Hirst inverse of P.

  The appearance of (i,K___) in the following list means that if P = X_i, then the K___ is the anticevian-associated cubic;

  (1, K323)

  (4, K780)

  (6, K128)

  (7, K623)

  (13, K419a)

  (14, K419b)

  (30, K472)

  (69, K779)

  (75, K766)

  (76, K738)

  (98, K777)

  (239, K770)

  (287, K776)

  (291, K769)

  (297, K718)

  (298, K867a)

  (299, K867b)

  (325, K778)

  (335, K768)

  (350, K767)

  (385, K739)

  (511, K357)

  (519, K1148)

  (694, K354)

  (1916, K322)

  (3978, K356)

  (11078, K859a)

  (11092, K859b)

  The appearance of (i,j) in the following list means that the X₂-Hirst inverse of X_i is X_j: (1,239), (3,401), (4,297), (5,40853), (6,385), (7,9436), (8,3912), (9,10025), (10,6542), (13,11078), (14,11092), (15,40854), (16,40855), (20,441), (21,448), [and others]

  Note that X₉₉ and X₁₉₀ are self-X₂-Hirst inverse. In general, a point P is self-X-Hirst inverse if and only if X lies on the circumconic with perspector X; thus, X is self-X₂-Hirst inverse if and only if X is on the Steiner circumellipse.

• X₄₀₈₉₆–X₄₀₉₀₈ — Points associated with paratriangles

  contributed by Clark Kimberling and Peter Moses, January 1, 2021. Let P =\(p : q : r\) (barycentrics) be a point in the plane of a triangle ABC. As in the preamble just before X₄₀₈₄₃, the paratriangle of P is the triangle A'B'C' with vertices given by

  A' =\(p : r : q\)

  B' =\(r : q : p\)

  C' =\(q : p : r\)

  For fixed P with paratriangle A'B'C', the locus of X such that the anticevian triangle of X is perspective to A'B'C' is the anticevian-associated cubic given by

  p (p r - q^2) y z^2 - p (p q - r^2) y^2 z + (cyclic) = 0,

  which is the cubic pK(P,P'), where P' = p^2 - q r : q^2 - r p : r^2 - p q is the X₂-Hirst inverse of P.

  The appearance of (i,K___) in the following list means that if P = X_i, then the K___ is the anticevian-associated cubic;

  (1, K323)

  (4, K780)

  (6, K128)

  (7, K623)

  [and others]

  The appearance of (i,j) in the following list means that the X₂-Hirst inverse of X_i is X_j: (1,239), (3,401), (4,297), (5,40853), (6,385), (7,9436), (8,3912), (9,10025), (10,6542), (13,11078), (14,11092), (15,40854), (16,40855), (20,441), (21,448), (22,15013), (23,40856), (25,15014), (27,447), (30,2), [and others]

  Note that X₉₉ and X₁₉₀ are self-X₂-Hirst inverse. In general, a point P is self-X-Hirst inverse if and only if X lies on the circumconic with perspector X; thus, X is self-X₂-Hirst inverse if and only if X is on the Steiner circumellipse.

• X₄₀₉₁₁–X₄₀₉₂₀ — Points associated the special dilation triangle

  contributed by Clark Kimberling and Peter Moses, January 4, 2021. Let T denote the pedal triangle of X₂. There is exactly one dilation of T from X₂ that is perspective to ABC. It is the (-1)-dilation, here named the special dilation triangle, with vertices given by

  A' = 4 a^2 : a^2 - b^2 + c^2 : a^2 + b^2 - c^2

  B' = b^2 + c^2 - a^2 : 4 b^2 : b^2 - c^2 + a^2

  C' = c^2 - a^2 + b^2 : c^2 + a^2 - b^2 : 4 c^2.

  See Clark Kimberling and Peter J. C. Moses, “Dilation-Induced Perspectivities among Triangles”, Journal for Geometry and Graphics 14 (2010) 1-14.

  X₂-of-A'B'C' = X₅₆₅₀, and X₃-of-A'B'C' = X₁₄₁

  The appearance of (T,i) in the following list means that A'B'C' is perspective to T, and the perspector is X_i:

  (ABC, 69)

  (tangential. 31521)

  (2nd Euler, 125)

  (4th extouch, 69)

  (Artzt, 2)

  [and others]

• X₄₀₉₂₁–X₄₀₉₂₆ — Perspectors involving the obverse triangle of X₆₉

  based on notes from Peter Moses, January 6, 2021. As in the preamble just before X₂₄₃₀₇, the obverse triangle, A'B'C', of a point P =\(p : q : r\) is given by

  A' =\(p : r : q\), B' =\(r : q : p\), C' =\(q : p : r\).

  In particular, for P = X₆₉,

  A' = b^2 + c^2 - a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2.

• X₄₁₀₁₆–X₄₁₀₇₁ — Altintas-isodynamic triangles

  contributed by César Eliud Lozada, January 11, 2021.

  Let ABC be a triangle. The parallel line to BC from P = X₁₅ (1st isodynamic point) intersects the circumcircle of ABC in A₁, A₂ and the Simson lines of A₁, A₂ cut at A'. Define B' and C' cyclically. Then A'B'C' is equilateral and its center lies on the Euler line of ABC. (Kadir Altintas, January 9, 2021)

  A' has barycentric coordinates:

  A' = -((S^2+SB*SC)*\(\sqrt{3}\)*S+(SA+3*SW)*S^2-2*(-SW*SA+S^2+SW^2)*SA)/(S^2-\(\sqrt{3}\)*S*SA-2*SW*SA) : SC : SB

  and squared-sidelength:

  L'² = S^2*\(\sqrt{3}\)/(3*S+\(\sqrt{3}\)*SW)
  When P=X₁₆ (2nd isodynamic point), the parallel line to BC from P intersect the circumcircle of ABC in two imaginary points, but the Simson lines of these points still intersect in a real point A″. The triangle A″B″C″ built in this way is also equilateral and:

  A″ = -((-S^2+SB*SC)*\(\sqrt{3}\)*S+(SA+3*SW)*S^2-2*(-SW*SA+S^2+SW^2)*SA)/(S^2-\(\sqrt{3}\)*S*SA-2*SW*SA) : SC : SB

  and its squared-sidelength is:

  L″² = S^2*\(\sqrt{3}\)/(-3*S+\(\sqrt{3}\)*SW)

  Triangles A'B'C' and A″B″C″ are referred here as 1st- and 2nd- Altintas-isodynamic triangles, respectively. A list of relations among these triangles and others can be seen here (X(41016).rtf).

• X₄₁₀₉₁–X₄₁₁₃₂ — Miscellaneous centers

  contributed by César Eliud Lozada, January 20, 2021. For definitions of triangles mentioned in these centers, see the index of triangles (IndexOfTrianglesReferencedInETC.html).

• X₄₁₁₃₃–X₄₁₁₅₄ — Inverses-in-permutation ellipses

  :contributed by Clark Kimberling and Peter Moses, January 25, 2021. Suppose that P =\(p : q : r\) (barycentrics) is a point in the plane of a triangle ABC. Suppose further that p,q,r are distinct homogeneous functions of a,b,c. The permutation ellipse of P, denoted by E(P), is introduced in the preamble just before X₃₄₃₄₁ as the ellipse that passes through the six points \(p : q : r\),\(q : r : p\),\(r : p : q\),\(p : r : q\),\(q : p : r\),\(r : q : p\), given by

  (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

  The centroid, X₂, is the center and the perspector of E(P).

  In C. Kimberling and P. Moses, ‘Permutation Ellipses’, Journal for Geometry and Graphics 24 (2020) 233-247, the following three special permutation ellipses are discussed: (1) trisector ellipse, (2) self-dual permutation ellipse; and (3) Steiner midway ellipse. Following are definitions and equations:

  1. The trisector ellipse passes through the points \(0:1:2\),\(0:2:1\),\(1:0:2\),\(2:0:1\),\(1:2:0\),\(2:1:0\). These six points trisect the sides of ABC. The ellipse is given by 2(x^2 + y^2 + z^2) - 5(y z + z x + x y) = 0.

  2. The unique self-dual permutation ellipse is given by x^2 + y^2 + z^2 - 4 (y z + z x + x y) = 0.

  3. The Steiner midway ellipse (SME), is, loosely speaking, the ellipse midway between the Steiner inellipse (SIE) and the Steiner circumellipse (SCE). That is, for each P on SCE, let P' be the intersection of the ray GP with SIE, and let P″ be the midpoint of PP'. Then SME is the set of all such midpoints. SME is given by 7(x^2 + y^2 + z^2) - 34(y z + z x + x y) = 0.

  In general, the inverse of a point P =\(p : q : r\) in an ellipse j (x^2 + y^2 + z^2) - k (y z + z x + x y) = 0 is the point k*p^2 - j*p*q + j*q^2 - j*p*r - k*q*r + j*r^2 : : .

  The appearance of (m,n) in the following list means that the trisector-ellipse-inverse of X(m) is X_n:

  (6,8859), (69,41133), (99,41134), (115,41135), (141,41136), (183,41137), (190,41138), (193,41139), (148,14971), (230,5032), (239,25055), (297,3545), (325,21356), (401,5054), (441,10304), (448,15671), (671,9166), (1635,24508), (3524,40884), (5055,40885), (5215,40871), (7840,21358), (8591,9167), (11078,22489), (11092,22490), (15699,40853), (17310,19875), (19883,40891), (22510,25165), (22511,25155), (37907,40856), (39663,39908)

  The appearance of (m,n) in the following list means that the self-dual-permutation-ellipse-inverse of X(m) is X_n:

  (1,41140), (8,41141), (37,41142), (39,41143), (75,41144), (98,41145), (183,41146)

  The appearance of (m,n) in the following list means that the midway-ellipse-inverse of X(m) is X_n:

  (115,41147), (148,41148), (230,41149), (239,41150), (287,41151), (297,3860), (325,41152), (385,41153), (671,41154), (441,15690), (15759,40884)

• X₄₁₁₅₅–X₄₁₁₆₁ — Inverses in circles

  based on notes from Predrag Terzic and Peter Moses, January 28-31, 2021. Terzic observed that the following 8 points lie on a circle, here named the Terzic circle:

  X₁₅, X₁₆, X₅₅, X₁₀₉, X₆₅₄, X₁₁₅₅, X₂₂₉₁, X₄₁₁₅₅.

  Moses found that the center of this circle is X₆₁₃₉, and also

  A-power = (b^2*c^2*(-a + b + c)*(-2*a^2 + a*b + b^2 + a*c - 2*b*c + c^2))/(2*(-a + b)*(a - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - b^2*c - a*c^2 - b*c^2 + c^3))

  squared radius = (a^2*b^2*c^2*(a^4 - a^3*b - a*b^3 + b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3 + c^4)^2)/(4*(a - b)^2*(a - c)^2*(b - c)^2*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - b^2*c - a*c^2 - b*c^2 + c^3)^2)

  See also the preamble just before X₄₁₁₆₃.

  The appearance of {\(n_{1}\), \(n_{2}\), …} in the following list means that the points \(X(n_{1})\), \(X(n_{2})\), … lie on a circle:

  {1,15,16,36,3465,4040,5526,5529}

  {2,15,16,23,110,111,352,353,5638,5639,6141,6142,7598,7599,7601,7602,7711,9138,9147,9153,9156,9157,9158,9162,9163,9212,9213,9978,9980,9998,9999,11199,11673,13114,13242,14660,14704,14705,32072,32073,32074,32526,33502,33503} (Parry circe)

  {4,15,16,186,3484,11674,13509,15412}

  {13,15,16,3165,5616,5669,6104,10658} {14,15,16,3166,5612,5668,6105,10657}

  {15,16,55,109,654,1155,2291,41155,41162,41163,41164,41165,41166,41167} (Terzic circle)

  {15,16,74,112,5667,9862,11587,40894,40895}

  {15,16,101,106,214,9321,11716,38013,38014,41183,41184,41185,41186,41187,41188,41189,41190,41191,41192,41193} (Moses isodynamic circle)

  {15,16,115,128,399,1263,1511,2079,10277,14367}

  {15,16,501,3743,5127,14838,14873,39149}

  {15,16,647,1495,14685,16319,35901}

  {15,16,667,1083,3230,11650,11651,11652}

  {15,16,1138,2132,6794,12112,14354}

  {15,16,5000,5001,6112,6113,6114,6115,6116,6117} (Moses radical circle)

  Every circle that passes through X₁₅ and X₁₆ has a center on the Lemoine Axis and is orthogonal to every circle ini the Schoute coaxal family, including these: circumcircle, Brocard circle, Lucas inner circle, Lucas circles radical circle, outer Montesdeoca-Lemoine circle, inner Montesdeoca-Lemoine circle.

• X₄₁₁₆₃–X₄₁₁₆₆ — Points on the Terzic circle

  based on notes from Peter Moses, January 30, 2021. Points X₄₁₁₆₃-X₄₁₁₆₅ were found using the following theorem, which was introduced in the preamble just before X₃₀₂₇:

  Suppose that O₁ and O₂ are circles. Let P₁ be a point on O₁, and let P₂ be the O₁-antipode of P₁. Let

  \(S_{i}\) = internal center of similitude of O₁ and O₂

  \(S_{e}\) = external center of similitude of O₁ and O₂

  Q₁ = P₁\(S_{i}\)∩P₂\(S_{e}\)

  Q₂ = P₁\(S_{e}\)∩P₂\(S_{i}\)

  Then Q₁ and Q₂ are a pair of antipodes on O₂, and the lines P₁P₂ and P₂Q₂ are parallel.

• X₄₁₂₀₂–X₄₁₂₂₄ — Christopher J. Bradley bicevian points and axes

  contributed by César Eliud Lozada, Februry 05, 2021. This section is based on the work On the Nine Intersections of Two Cevian Triangles (Article 20), by Christopher J. Bradley (with symmetrical notations by César Lozada).

  Let :g:`U <U>`, :g:`X <X>` be two distinct points on the plane of :g:`ABC << A B C >>`, neither on its sidelines nor on a line parallel to a sideline through the opposite vertex. Let A_uB_uC_u and A_xB_xC_x be the :term:`cevian triangles <cevian-triangle>` of :g:`U <U>` and X, respectively. We introduce the nine intersections of these two triangles:

  A₁ = B_uC_u ∩ B_xC_x, and cyclically B₁, C₁

  A₂ = A_uB_u ∩ A_xC_x, and cyclically B₂, C₂

  A₃ = A_xB_x ∩ A_uC_u, and cyclically B₃, C₃

  Then, assuming that U =\(u : v : w\) and X = x : y z (barycentrics):

  A, B₁, C₁ are collinear on a line \(a_{1}\); B, C₁, A₁ are collinear on a line \(b_{1}\); C, A₁, B₁ are collinear on a line \(c_{1}\). BC, B₂C₂, B_uC_x concrr in a point A'; CA, C₂A₂, C_uA_x concur in a point B'; AB, A₂B₂, A_uB_x concur in a point C'. BC, B₃C₃, B_xC_u concur in a point A″; CA, C₃A₃, C_xA_u concur in a point B″; AB, A₃B₃, A_xB_u concur in a point C″. A', B', C' are collinear on a line L' and also A″, B″, C″ are collinear on a line L″. Moreover, these two lines are concurrent with UX at a point O(U,X), here named the Bradley bicevian point of (U, X).

  Barycentric coordinates of this point are O(U, X) = u^2*y*z-x^2*v*w : : . In other words, O(U, X) is the U-Hirst inverse-of-X = X-Hirst inverse-of-U (see the ETC glossary (glossary.html)). A, A₂, A₃ are collinear, as are B, B₂, B₃ and C, C₂, C₃. The last result means that ABC, A₂B₂ and A₃B₃ are perspective by pairs with common perspector Q(U,X) = (v*z+w*y)*u*x ::. A₁B₁C₁ is perspective to these triangles: ABC, A_uB_uC_u, A_xB_xC_x, A₂B₂C₂ and A₃B₃C₃. Let Q₂(U,X) and Q₃(U,X) be the perspectors of A₁B₁C₁ with A₂B₂C₂ and A₃B₃C₃, respectively. Q(U,X), Q₂(U,X) and Q₃(U,X) lie on a line, here named the Bradley bicevian axis of (U, X), whose trilinear pole has barycentric coordinates:

  T(U, X) = (x*v^2*z-y^2*u*w)*(x^2*w^2-z^2*u^2)*(y*w^2*x-z^2*v*u)*(y^2*u^2-x^2*v^2) : :

  Note that A₁B₁C₁ is the side-triangle of the U- and X-cevian triangles. For shortening, the term side-of-cevians-of-(U,X) will be used here for this triangle.

  The appearance of (i, j, k) in the following list means that the Bradley bicevian point of (X_i, X_j) is X_k:

  (1, 2, 239), (1, 3, 1936), (1, 4, 243), (1, 5, 2596), (1, 6, 238), (2, 3, 401), (2, 4, 297), (2, 5, 40853), (2, 6, 385), (3, 4, 450), (3, 5, 41202), (3, 6, 511), (4, 5, 41203), (4, 6, 41204), (5, 6, 41205)
  The appearance of (i, j, k) in the following list means that the trilinear pole of the Bradley bicevian axis of (X_i, X_j) is X_k:

  (1, 2, 4589), (1, 3, 41206), (1, 4, 41207), (1, 6, 4584), (2, 3, 41208), (2, 4, 22456), (2, 6, 41209), (3, 6, 2966), (4, 6, 41210)

  The appearance of (i, j, m, n) in the following list means that the perspectors of the side-cevians-of-(X_i, X_j) with the cevian-of-X_i and the cevian-of-X_j are X(m) and X_n, respectively:

  (1, 2, 244, 1015), (1, 3, 2638, 3270), (1, 4, 2310, 3270), (1, 5, 41211, 41218), (1, 6, 3248, 1015), (1, 7, 2310, 3022), (1, 8, 2310, 3271), (2, 3, 35071, 2972), (2, 4, 115, 125), (2, 5, 39019, 35442), (2, 6, 1084, 3124), (2, 7, 1086, 11), (2, 8, 1146, 11), (3, 4, 20975, 34980), (3, 5, 41212, 41219), (3, 6, 20975, 3269), (3, 7, 41214, 15616), (3, 8, 41215, 41220), (4, 5, 24862, 41221), (4, 6, 34980, 3269), (4, 7, 3270, 3022), (4, 8, 3270, 3271), (5, 6, 41213, 41222), (5, 7, 41216, 31889), (5, 8, 41217, 41223), (6, 7, 35505, 15615), (6, 8, 35506, 41224), (7, 8, 3022, 3271)

• X₄₁₂₃₁–X₄₁₂₇₈ — Gibert quadratic transformations and inverses

  based on notes from Bernard Gibert, and related points contributed by Peter Moses, February 11, 2021. Let X \(x : y : z\) (barycentrics) be a point in the plane of a triangle ABC. Define

  \(F(X) = b^{4} c^{2} x^{2}-b^{2} c^{4} x^{2}-a^{4} c^{2} x y+b^{2} c^{4} x y+a^{4} b^{2} x z-b^{4} c^{2} x z-a^{4} b^{2} y z+a^{4} c^{2} y z\): : ,

  with inverse given by

  invF(X) = a^2 (b^4 x^2-c^4 x^2-a^2 b^2 x y+c^4 x y-b^4 x z+a^2 c^2 x z+a^2 b^2 y z-a^2 c^2 y z) : :

  In the sequel, these transformations are denoted by BGF and invBGF.

  The singular points of BGF are X₂, X₆, and X₃₂. If X is on X₂X₆, then F(X) = X₆. If X is on X₂X₃₂, then F(X) = X₈₃. If X is on X₆X₃₂, then F(X) = X₂.

  The fixed points of F; i.e., points X such that BG(X) = invBG(X) = X, are the points on the circumconic having perspector X₆₆₉.

  BGF maps every line through X₂ onto itself; in particular, BGF(Euler line) = Euler line.

  BGF maps the circumconic having perspector X₅₁₂ onto the Kiepert hyperbola.

  BGF maps the circumconic having perspector X₆₈₈ onto the circumconic having perspector X₅₁₂.

  The transformations BGF and invBGF have many connections among pk cubics. In order to state a few of them, certain notations will be used: “/” denotes barycentric division, and “*” denotes barycentric product, and if X is a point, then gX = isogonal conjugate of X, and tX = isotomic conjugate of X.

  Let pk1 = pK(Ω,P) be a pK cubic and let pk2 = pK(Ω’,P') be another pK cubic such that the isopivot of Q' is Ω’/P'.

  1. The cubics pk1 and pk2 meet the line at infinity at the same points if and only if Q' lies on a psK cubic that is a pK cubic if and only if the points X₂, Ω, and P are collinear.

  2. The cubics pk1 and pk2 meet the circumcircle at the same points if and only if the points X₆, Ω/X₆, and P are collinear; or, equivalently, the points X₃₂, Ω, and P*X₆ are collinear. Note that Ω/X₆ = tgΩ and P*X₆ = gtΩ.

  (3) Conditions (1) and (2) hold simultaneously if and only if

  P lies on the lines X₂Ω and X₆tgΩ meeting at BGF(Ω) and

  Ω lies on the lines X₂ P and X₃₂ gtP meeting at invBGF(P).

• X₄₁₂₇₉–X₄₁₂₉₂ — 2nd Lozada perspectors

  contributed by Vu Thanh Tung, February 14, 2021. As a sequel to the preamble just before X₆₀₅₆, in which Lozada perspectors are introduced, this section is based on the following definition: Let A₁B₁C₁ be the cevian triangle of a point P =\(p : q : r\) (barycentrics) in the plane of a triangle ABC. Let L_A be the line, other than BC, that passes through A₁ and is tangent to the A-excircle. Let A' be the touch point, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector, here named the 2nd Lozada perspector of P, denoted by V(P), is given by

  V(P) = (a + b - c)(a - b + c) p^2 : (b + c - a)(b - c + a) q^2 : (c + a - b)(c - a + b) r^2.

  See 2nd Lozada Perspector (2ndLozadaPerspector.pdf).

  The appearance of (i,j) in the following list means that V(X_i) = X_j:

  (1,56), (2,7), (3,7335), (5,41279), (6,1397), (7,479), (8,8), (9,55), (10,12), (31,41280), (32,41281), (36,41282), (75,6063), (76,41283), (83,41284), (141,41285), (560,41286), (561,41287), (1501,41288), (1502, 41289), (1928,41290), (2321,6057), (2887,41291), (3676,451292)

• X₄₁₃₀₁–X₄₁₃₀₈ — Points associated with Steiner parabolas

  contributed by César Eliud Lozada, February 21, 2021. Let ABC be a triangle and P a point on its plane. Let A_b, A_c be the points at which the parallel line to BC through P cuts AC and AB, respectively and denote B_c, B_a, C_a, C_b cyclically. It is well known that these six points lie on a conic and also that this conic is a parabola SP(P) if and only if P lies on the Steiner inellipse of ABC. This parabola is here named the Steiner parabola of P.

  The following algebraic method for calculating the focus F(P) of the Steiner parabola SP(P) follows from the preambles just before X₃₁₆₄₄ and X₃₈₀₁₇:

  Let X = 3*PG/5, where G=X₂, and let X' = barycentric product of P and X. Let Y = X₂-Ceva conjugate-of-P. Since P lies on the Steiner inellipse, Y lies on the line at infinity. Let A'B'C' be the orthic triangle of the cevian triangle of Y. Then A'B'C' and ABC are perspective. Denote their perspector by Y'. The focus F(P) is the midpoint of X' and Y'. The axis of SP(P) passes through Y.

• X₄₁₃₈₀–X₄₁₃₈₈ — Centers of antiparallels conics

  contributed by César Eliud Lozada, February 22, 2021. Draw antiparallels through the symmedian point :g:`K <K>`. The points where these lines intersect the sides then lie on a circle, known as the cosine circle (or sometimes the second Lemoine circle). (Reference: Weisstein, Eric W. “Cosine Circle.” From MathWorld–A Wolfram Web Resource). More generally, if P is a point on the Jerabek circumhyperbola of a triangle ABC then the antiparallel lines through P cut the sidelines of the triangle in six points lying on a conic, here named the antiparallels conic of P and denoted as 𝕁(P).

  The appearance of (i, j) in the following list means that the center of 𝕁(X_i) is X_j:

  (3, 37864), (4, 6), (6, 6), (54, 41380), (64, 33584), (65, 41381), (66, 41382), (67, 41383), (68, 41384), (69, 41385), (70, 41386), (71, 41387), (72, 41388), (74, 40355).

  𝕁(P) is a circle for P=X₆ and a degenerated conic for P in {X₁₉₈₇, X₈₆₇₈, X₈₆₇₉}. 𝕁(P) is never a parabola.

• X₄₁₃₉₄–X₄₁₄₃₀ — Perspectors involving g-triangles

  contributed by Clark Kimberling and Peter Moses, February 25, 2021. Mappings g and h are defined in the preamble just before X₃₃₆₂₈ as follows. Suppose that P =\(p : q : r\) is a point. The points

  q - r : r - p : p - q and 2p - q - r : 2q - r - p : 2r - p - q

  clearly lie on the line at infinity, so that their isogonal conjugates,

  \(g(P) = a^{2}/(q-r)\): b^2/(r-p) : c^2/(p-q) and \(h(P) = a^{2}/(2p-q-r)\): b^2/(2q-r-p) : c^2/(2r-p-q),

  lie on the circumcircle. Thus, if T is a central triangle, then g(T) and h(T) are central triangles inscribed in the circumcircle.

  For many choices of triangle T, there are triangles T' such that T is perspective to g(T'). This section lists associated perspectors.

  The appearance of (T,T',i) in the following list means that T is perspective to g(T') and the perspector is X_i:

  (ABC, 3rd Brocard, 32)

  (ABC, orthocentroidal, 6)

  (ABC, 2nd Parry, 187

  (ABC, centers of the Apollonius circles, 32)

  (ABC, Gemini 19, 649

  (medial, MacBeath, 4)

  (anticomplementary, Steiner, 99)

  [and others]

  Following are two examples of pairs, (T,T') of named triangles T and T' such that T = g(T'):

  orthocentroidal triangle = g(cirum-symmedial triangle)

  1st cirumperp triangle = \(g(Gemini 7) = g(Gemini 15)\)= \(g(Gemini 17) = g(Gemini 25)\)

  The appearance of T in the following list means that T is perspective to \(g(T) = 1st cirumperp triangle\), and the perspector is X₃: medial, tangential, 2nd circumperp, inner Napoleon, outer Napoleon, inner Fermat, outer Fermat, inner Vecten, outer Vecten, 1st Neuberg, 2nd Neuberg, Fuhrmann, 1st Brocard, Kosnita, McCay, Trihn, reflection of ABC in X₅, Ara, 2nd Euler

  The appearance of (T,i) in the following list means that T is perspectiive to \(g(T) = 1st cirumperp triangle\), and the perspector is X_i:

  (3rd Euler, 2), (4th Euler, 4), (intouch, 55), (hexyl, 40), (Yff central, 7589), (inner tangential mid-arc, 8075), (tangential of 1st circumperp, 11495), (tangential of 2nd circumperp, 12513), (1st Sharygin, 4220), (2nd Sharygin, 105), (Honsberger, 7676), (2nd Pamfilos-Zhou, 8224), (2nd extouch, 7580), (3rd mixtilinear, 1), (4th mixtilinear, 165), (6th mixtilinear, 165), (outer tangential mid-arc, 8076), (1st Conway, 7411), (incircle-inverse of ABC, 57), (inner Hutson, 8107), (outer Hutson, 8108), (T(-2,1) in TCCT, Art. 6.41, 9), (T(-1,3) in TCCT, Art. 6.41, 7991), (Hutson intouch, 56), (1st EhrmannT, 165), (Atik, 10860),

  The appearance of (T,i) in the following list means that T is perspective to g(T), and the persepctor is X_i:

  (tangential,1627), (3rd Brocard, 32), (MacBeath, 25), (Thomson, 6), (anti-orthocentroidal, 110), (Wasat, 1691)

  The locus of a point X such that for T = anticevian triangle of X, we have \(g(t) = T is the cubic K141\); see Bernard Gibert, K141: pK(X₂,X₇₆).

  The locus of a point X such that for T = cevian triangle of X, we have g(t) = T is the cubic pK(X₃₀₈, X₃₈₀), which passes through X_i for these i: 2, 76, 83, 264, 308, 1799, 17907.

  See César Lozada, Perspectivities involving the g and h mappings (g(T)_h(T).pdf).

• X₄₁₄₃₁–X₄₁₄₇₉ — Perspectors involving g-triangles

  contributed by Clark Kimberling and Peter Moses, February 28, 2021. Mappings g and h are defined in the preamble just before X₃₃₆₂₈ as follows. Suppose that P =\(p : q : r\) is a point. The points

  q - r : r - p : p - q and 2p - q - r : 2q - r - p : 2r - p - q

  clearly lie on the line at infinity, so that their isogonal conjugates,

  \(g(P) = a^{2}/(q-r)\): b^2/(r-p) : c^2/(p-q) and \(h(P) = a^{2}/(2p-q-r)\): b^2/(2q-r-p) : c^2/(2r-p-q),

  lie on the circumcircle. Thus, if T is a central triangle, then g(T) and h(T) are central triangles inscribed in the circumcircle.

  For many choices of triangle T, there are triangles T' such that T is perspective to h(T'). This section lists associated perspectors.

  The appearance of (T,T',i) in the following list means that T is perspective to h(T') and the perspector is X_i:

  See César Lozada, Perspectivities involving the g and h mappings (g(T)_h(T).pdf).

• X₄₁₅₃₇–X₄₁₆₇₉ — Pairs of orthologic pedal triangles

  contributed by César Eliud Lozada, March 07, 2021. Let :g:`P’ <P’>`, :g:`P″ <P″>` be any two distinct, finite points not both on the circumcircle of a triangle :g:`ABC << A B C >>` and such that they are collinear with its circumcenter :g:`O <O>`=:ref:`X<sub>3</sub> <X<sub>3</sub>>`. Then the :term:`pedal triangles <pedal-triangle>` :g:`A’B’C’ << A’ B’ C’ >>` and :g:`A″B″C″ << A″ B″ C″ >>` of :g:`P’ <P’>` and :g:`P″ <P″>` are orthologic.

  Let Q' and Q″ be the orthologic centers (A'B'C' to A″B″C″) and (A″B″C″ to A'B'C'), respectively. If P' is fixed and P″ moves along the line OP', then Q' describes a rectangular hyperbola ℍ(P') circumscribed to A'B'C', whilst Q″ moves on a line 𝕃(P'). ℍ(P) is named here the bipedal circumcentral conic of P’ (shortened to the BPC-conic of P’) and 𝕃(P') is named here the bipedal circumcentral line of P’, abbreviated as the BPC-line of P’. If P' =\(x:y:z\) (barycentrics), then

  the center of ℍ(P') is:

  ℍ*(P') = (a^2*b^4*c^4*x^3+((3*SA-SW)*S^2-2*(SB+SC)^2*SA)*b^2*c^2*x*y*z+((S^2-SB^2)*b^2*z+(S^2-SC^2)*c^2*y)*SA*a^2*y*z-b^2*c^2*((2*S^2-(2*SB+SC)*SB)*b^2*z+(2*S^2-(SB+2*SC)*SC)*c^2*y)*x^2+((S^2*SA+SB^3)*b^4*z^2+(S^2*SA+SC^3)*c^4*y^2)*x)*(SB+SC) : : the tripole of 𝕃(P') is:

  𝕃*(P') = (SA*a^2*y-SB*b^2*x)*(SA*a^2*z-SC*c^2*x)*(b^2*z+c^2*y) : :

  If P' lies on K174, 𝕃, then *(P') lies on the circumcircle of ABC.

  The appearance of (i, j, m, n) in the following lists means that the orthological centers of the pedal triangles of X_i and X_j are X(m) and X_n: (1, 3, 1, 10), (1, 35, 3649, 12), [and others]. The appearance of (i, j) in the following list means that the center of the BPC-conic of X_i is X_j:

  The appearance of (i, j) in the following list means that the tripole of the BPC-line of X_i is X_j: (1, 4552), (2, 41676), (4, 14570), (5, 41677), (6, 99), [and others].

• X₄₁₆₈₄–X₄₁₇₅₆ — Pairs of orthologic reflection triangles

  contributed by César Eliud Lozada, March 08, 2021. If P is point on the plane of ABC, the reflections of P in the sidelines of ABC are the vertices of a new triangle named the reflection triangle of PLet :g:`P’ <P’>`, :g:`P″ <P″>` be any two distinct points on the plane of a triangle :g:`ABC << A B C >>` and such that they are collinear with its circumcenter :g:`O <O>`=:ref:`X<sub>3</sub> <X<sub>3</sub>>`. Then the :term:`reflection triangles <reflection-triangle>` :g:`A’B’C’ << A’ B’ C’ >>` and :g:`A″B″C″ << A″ B″ C″ >>` of :g:`P’ <P’>` and :g:`P″ <P″>` are orthologic.

  Let Q' and Q″ be the orthologic centers (A'B'C' to A″B″C″) and (A″B″C″ to A'B'C'), respectively. If P' is fixed and P″ moves along the line OP', then Q' describes a rectangular hyperbola ℍ(P') circumscribed to A'B'C', whilst Q″ moves on a line 𝕃(P'). ℍ(P') is named here the bireflection circumcentral conic of P’ (shortened to the BRC-conic of P’) and 𝕃(P') is named here the bireflection circumcentral line of P’, abbreviated as the BRC-line of P’.

  If P' =\(x : y : z\) (barycentrics). The center of ℍ(P') is:

  ℍ*(P') = (a^2*b^2*c^2*x - (S^2 - SB^2)*b^2*z - (S^2 - SC^2)*c^2*y)*(b^2*c^2*x^2 - 2*SA*a^2*y*z - (SA - SB)*b^2*x*z - (SA - SC)*c^2*x*y)*a^2 : :

  ℍ*(P') coincides with the Hatzipolakis-Moses nine-point image of P', denoted by HM(P'), defined in X₁₂₈₂₄. The tripole of 𝕃(P') is:

  𝕃*(P') = b^2*c^2*(b^2*SB*x - a^2*SA*y)*(c^2*SC*x - a^2*SA*z)*

  (2*a^2*c^2*SB*y^2 + c^2*(S^2 - (SB - 2*SC)*SB)*x*y + b^2*(S^2 - 3*SB^2)*x*z + a^2*(S^2 + (2*SA - SB)*SB)*y*z)*

  (2*a^2*b^2*SC*z^2 + b^2*(S^2 - (SC - 2*SB)*SC)*x*z + c^2*(S^2 - 3*SC^2)*x*y + a^2*(S^2 + (2*SA - SC)*SC)*y*z) : :

  Note: The orthologic center (reflection-of-P' to reflection-of-P″) is the reflection in P' of the orthologic center (pedal-of-P' to pedal-of-P″). (See X₄₁₅₃₇)

  The appearance of (i, j, m, n) in the following lists means that the orthological centers of the reflection triangles of X_i and X_j are X(m) and X_n: (1, 3, 1, 355), (1, 35, 79, 3585), (1, 36, 7972, 41684), (1, 40, 15071, 5693), [and others].

  The appearance of (i, j) in the following list means that the center of the BRC-conic of X_i is X_j: {1, 11570}, {2, 12824}, {4, 1986}, {5, 11557}, {6, 5477}, {15, 6783}, {16, 6782}, [and others].

  The appearance of (i, j) in the following list means that the tripole of the BRC-line of X_i is X_j: (2, 6331), (4, 30450), (5, 38342), [and others].

• X₄₁₇₅₉–X₄₁₇₈₀ — Points on the cubic K1188

  contributed by Peter Moses, March 9, 2021.

• X₄₁₇₈₅–X₄₁₇₉₁ — Points on the cubic K1189

  contributed by Peter Moses, March 9, 2021.

• X₄₁₈₉₀–X₄₁₉₁₁ — Centers from some problems in Russian Sharygin Olympiads

  contributed by César Eliud Lozada, March 12, 2021.

• X₄₁₉₃₂–X₄₁₉₃₇ — Points on the square of the circumcircle

  contributed by Clark Kimberling and Peter Moses, March 13, 2021. The square of the circumcircle is given by the barycentric equation

  a^8 y^2 z^2 - 2 b^4 c^4 x^2 y*z + (cyclic) = 0.

  The appearance of (i,j) in the following list means that X_i is on the circumcircle, and X_i^2 = X_j: (74,40353), (98, 41932), (99,4590), (100,1252), (101,23990), (104,41933), (105, 41934), (106, 31935), (107,23590), (108,23985), (109,23979), (110,23357), (111, 41936), (112, 41937)(476,23588), (934,23971)

  The cube of the circumcircle is given by

  a^18 y^3 z^3 + 3 a^12 x y^2 z^2 (c^6 y + b^6 z) - 7 a^6 b^6 c^6 x^2 y^2 z^2 + (cyclic) = 0.

• X₄₁₉₄₃–X₄₁₉₈₀ — Points on the Evans conic

  contributed by Peter Moses, March 14-16, 2021. For a discussion of the Evans conic, see Evans Conic. If k1 and k2 are functions symmetric in a,b,c and homogeneous of degree 0, then the points

  E₁(k1,k2) = 3*a^2*(a^2 - b^2 - c^2)*k2 - 4*k1*S^2 - 2*a^2*\(\sqrt{k1^{2} + 2 k1 k2 + 3 k2^{2}}\)*S : : and

  E₂(k1,k2) = 3*a^2*(a^2 - b^2 - c^2)*k2 - 4*k1*S^2 + 2*a^2*\(\sqrt{k1^{2} + 2 k1 k2 + 3 k2^{2}}\)*S : :

  lie on the Evans conic. The lines tangent to the Evans conic at the two points meet on the Euler line in the point T(k1,k2) = (k1 + 3*k2)*X₂ - (k1 + k2)*X₃.

  The appearance of (k1,k2) → (i,j); h in the following list means that E₁(k1,k2) = X_i, E₂(k1,k2) = X_j, and T(k1,k2) = X(h): ,br> (-2,1) → (14,13); 549

  (0, 1) → (15,16); 5

  (-6,1) → (18,17); 550

  (-3,2) → (3071,3070); 140

  , [and others].

  The points E₁ and E₂ defined above are also Gibert points, as defined in the preamble just before X₄₂₀₈₅; specifically, in the notation of that preamble,

  E₁(k1,k2) = (\(\sqrt{3 (k1^{2} + 2 k1 k2 + 3 k2^{2})}\) , k1, 2*k1 + 3*k2) Gibert point;

  E₂(k1,k2) = (-\(\sqrt{3 (k1^{2} + 2 k1 k2 + 3 k2^{2})}\), k1, 2*k1 + 3*k2) Gibert point.}

• X₄₁₉₉₃–X₄₂₀₀₄ — Points on Simmons inconics

  contributed by Peter Moses, March 17-18, 2021. See Bernard Gibert, Simmons Conics. In this section, the Simmons inconic with foci X₁₃ and X₁₅ is the 1st Simmons inconic and the Simmons inconic with foci X₁₄ and (16) is the 2nd Simmons inconic.

  Suppose that IC is an inconic, and let P be its perspector. Then the barycentric product P*(Steiner inellipse) is on IC. In particular, if U is on the line at infinity, then the barycentric product X₁₃*U^2 is on the 1st Simmons inconic, and X₁₄*U^2 is on the 2nd Simmons inconic.

Part 22

• X₄₂₀₀₅–X₄₂₀₆₆ — Perpsectors associated with inverse triangles

  contributed by Clark Kimberling and Peter Moses, March 18, 2021. Suppose that A'B'C' is a triangle with vertices A', B', C' represented in normalized barycentric coordinates. Let M be the matrix representation of A'B'C', and let inv(M) denote the inverse of M. Then the rows of inv(M), interpreted as vertices of a triangle, define the inverse triangle of A'B'C'. The row sums of the inverse triangle are all 1, so that this triangle is “automatically” normalized.

  Let P =\(p : q : r\) be a point, and let A'B'C' be the cevian triangle of P, given by A ‘ =\(0 : q : r\), with B' and C' determined cycllically from A'. The inverse triangle of A'B'C' is the triangle A″B″C″ given by A″ = - q - r : q + r : p + q, with B″ and C″ determined cyclically from A″; thus, A″B″C″ is the anticevian triangle of the point q + r : r + p : p + q.

  Here, let A'B'C' be the cevian triangle of P. The inverse of A'B'C' is the triangle A″B″C″ given by A″ = 0 : p - q + r : p + q - r, with B″ and C″ determined cyclically from A″; thus A″B″C″ is the cevian triangle of the point - p + q + r ; p - q + r : p + q - r.

  A triangle A'B'C' is inscribed in ABC if and only if its inverse circumscribes ABC.

  See the preambles just before X₄₃₂₈₀, X₄₃₃₄₄, and X₄₆₉₆₁.

• X₄₂₀₇₄–X₄₂₀₈₄ — Points on the Hofstadter inellipse

  contributed by Peter Moses, March 20, 2021. The Hofstadter inellipse is introduced at X₃₅₉, where it is denoted by E(1/2).

  The Hofstadter inellipse is also the incentral inellipse, the trilinear square of the antiorthic axis, the X₁-Ceva conjugate of the antiorthic axis, the barycentric product X₁*[Steiner inellipse], the barycentric product X₉*[incircle], and the locus of trilinear poles, wrt the incentral triangle, of lines passing through X₁. (Randy Hutson, May 31, 2021)

• X₄₂₀₈₅–X₄₂₂₈₄ — Gibert (i,j,k) points

  contributed by Peter Moses, March 22, 2021. Bernard Gibert has noted a set of points in connecton with the cubic K1191. These points are given by the combo

  SW*X₆*i + \(\sqrt{3}\)*S*(j*X₄ + k*X₃) where (i,j,k) are constants or other 0-degree functions of a,b,c. See K1191.

• X₄₂₂₈₅–X₄₂₂₈₆ — Perpsectors associated with product triangles

  contributed by Clark Kimberling and Peter Moses, March 23, 2021. Let T₁ = A₁B₁C₁ be a triangle, and let M₁ be the matrix whose rows are the normalized barycentrics of A₁, B₁, C₁, respectively.

  Let T₂ = A₂B₂C₂ be a triangle, and let M₂ be the matrix whose rows are the normalized barycentrics of A₂, B₂, C₂, respectively.

  Let M be the matrix sum M₁ + M₂. The triangle sum T₁ + T₂ is here defined as the triangle whose vertices are given by the rows of M₁ + M₂. (Note that triangle sum is non-associative and that there is no additive identity, hence no additive inverses.)

  In this section “cevian(P)” means the cevian triangle of P, and “anticevian U″ means the anticevian triangle of U.

  cevian(P)+anticevian(U) is perspective to both cevian(P) and anticevian(U), and the perspector is the P-Ceva conjugate of U, given by

  u (-u/p + v/q + w/r) : v (-v/q + w/r + u/p) w (-w/r + u/p + v/q).

  If P = X₁, then cevian(P) + anticevian(U) is perspective to ABC for U on a certain cubic through X₁, X₁₀, and X₁₁₂₅. If U = X₁, the perspector is X₄₂₂₈₅; if U = X₁₁₂₅, the perspector is X₅₅₁.

  If P = X₆, then cevian(P) + anticevian(U) is perspective to ABC for U on a certain cubic through X₁, X₁₄₁, and X₃₅₈₉. If U = X₁₄₁, the perspector is X₄₂₂₈₆.

• X₄₂₂₈₇–X₄₂₄₁₀ — Polarologic and Polelogic centers & T-isogonal-axes

  contributed by César Eliud Lozada, March 24, 2021. Let T ‘ = A₁B₁C₁ and T″ = A₂B₂C₂ be two distinct scalene triangles. Let (a’), (b’), (c’) be the :term:`trilinear polars <trilinear-polar>` of the vertices of T ‘ with respect to :g:`T″ <T″>` and let A', B', C' be the :term:`trilinear poles <trilinear-pole>` of the sidelines of T ‘ with respect to :g:`T″ <T″>`. Inversely, let (a”), (b”), (c”) be the trilinear polars of the vertices of :g:`T″ <T″>` with respect to T ‘ and A″, B″, C″ the trilinear poles of the sidelines of :g:`T″ <T″>` with respect to T ‘. Then:

  If (a’), (b’), (c’) concur in a point :g:`P’ <P’>` then (a”), (b”), (c”) also concur in a point :g:`P″ <P″>`. If A', B', C' are collinear on a line ℓ’ then A″, B″, C″ are also collinear on a line ℓ”. If T ‘ and :g:`T″ <T″>` satisfy one of the properties (1) or (2), then they satisfy the other property.

  T ‘ and T″ have the concurrences in (1) and the collinearities in (2) if the isogonal conjugates of the vertices of T ‘ with respect to T″ are collinear on a line 𝓂’. In this case, the isogonal conjugates of the vertices of T″ with respect to T ‘ are also collinear on a line 𝓂”.

  Here, points P' and P″ are named the polarologic centers of (T ‘ to T″) and (T″ to T ‘) and the tripoles of lines ℓ’ and ℓ” are referred as the polelogic centers of (T ‘ to T″) and (T″ to T ‘) . Also, the lines 𝓂’ and 𝓂” are introduced as the T-isogonal-axes of (T ‘ wrt T″) and (T″ wrt T ‘).

  The appearance of (T, [i, j], [m, n]) in the following list means that triangles ABC and T have polarologic centers X_i, X_j and polelogic centers X(m), X_n: (ABC-X3 reflections, [1350, 6], [42287, 69]), (1st anti-circumperp, [20477, 6], [30441, 670]), (anti-Honsberger, [182, 32], [42288, 83]), [and others].

  The appearance of (T, [i, j], [m, n]) in the following list means that triangles anticomplementary and T have polarologic centers X_i, X_j and polelogic centers X(m), X_n: (anti-Euler, [42329, 264], [42330, 95]), (3rd anti-Euler, [1510, 42331], [42332, 42333]), (Aquila, [42334, 86], [42335, 1268]), [and others].

  The appearance of (T, [i, j], [m, n]) in the following list means that triangles MEDIAL and T have polarologic centers X_i, X_j and polelogic centers X(m), X_n: (anti-Aquila, [15569, 3739], [39721, 7]), (Euler, [5480, 141], [42352, 253]), (2nd Euler, [42353, 141], [42354, 42355]), (3rd Euler, [42356, 141], [42357, 190]), [and others]

  The appearance of (T, [i, j], [m, n]) in the following list means that triangles ORTHIC and T have polarologic centers X_i, X_j and polelogic centers X(m), X_n: (anti-excenters-reflections, [36990, 6], [42373, 393]), (Euler, [5480, 53], [42374, 6]), (2nd Euler, [42353, 53], [42375, 42376]), (3rd Euler, [42356, 53], [–, –]), (4th Euler, [3826, 53], [–, –]), [and others]

  The appearance of (T, i, j) in the following list means that the tripoles of the T-isogonal-axes of triangles ABC and T are X_i and X_j: (ABC-X3 reflections, 2, 2), (1st anti-circumperp, 2, 2), (anti-Honsberger, 3407, 76), [and others].

• X₄₂₄₁₃–X₄₂₄₂₁ — Points on the Cullen cubic

  contributed by Peter Moses, March 25, 2021. The Cullen cubic is indexed as K369; see K369. Points (42413)-X₄₂₄₂₀ are Gibert points, as in the preamble just before X₄₂₀₈₅.

• X₄₂₄₂₂–X₄₂₄₂₆ — Points on the nine-point circle

  contributed by Clark Kimberling and Peter Moses, March 26, 2021. The appearance of {i,j} in the following list means that X_i and X_j are a pair of antipodes on the nine-point circle; i.e., each is the reflection of the other in the center, X₅, of the nine-point circle: {11,119}, {113,125}, {114,115}, {115,114}, {116,118}, {117,124}, [and others].

  Theorem (Moses): Suppose that P =\(p : q : r\) is a point, and define \(f(P) = p ((a^{2} - b^{2} + c^{2}) q + (-a^{2} - b^{2} + c^{2}) r) \cdot ((-b^{2} + c^{2}) q r + p (c^{2} q - b^{2} r))\): : . The point f(P) is on the nine-point circle, and f(P) is the center of the rectangular circumhyperbola {{:g:`A <A>`, :g:`B <B>`, :g:`C <C>`, :ref:`X<sub>4</sub> <X<sub>4</sub>>`, P}}. The perspector of the hyperbola lies on the orthic axis.

  The appearance of {i, {i(1),i(2),…i(n),} {name}} in the following list means that X_i = f(X_k) for k = 1,2,…,n, and that the circumhyperbola passing through these points has the indicated name, if there is one: [list omitted here].

• X₄₂₄₂₉–X₄₂₄₃₆ — Gibert points on 7th Evans cubic, K1196

  contributed by Peter Moses, March 29, 2021. See K1196.

• X₄₂₄₃₇–X₄₂₄₆₃ — Products X_i*T for selected triangles T

  contributed by Clark Kimberling and Peter Moses, March 31, 2021. If X is a normalized triangle center and T a normalized central triangle, then the left-product X*T is a triangle center. Let P =\(p : q : r\) be a point. Centers X₄₂₄₃₇-X₄₂₄₅₀ are products X_i*(cevian triangle of P), given by

  p (q + r) (p y + q y + p z + r z) : q (p + r) (p x + q x + q z + r z) : (p + q) r (p x + r x + q y + r y).

  Centers X₄₂₄₅₁-X₄₂₄₆₃ are products X_i*(anticevian triangle of P), given by p*((p + q - r)*(p - q + r)*x + (p - q - r)*(p + q - r)*y + (p - q - r)*(p - q + r)*z) : :

  The appearance of (i,j,k) in the following list means that X_i*(cevian triangle of X_j = X_k (1,1,2292), (1,2,10), (1,4,65), (1,6,20969), (1,7,1), (1,8,8), (1,20,5930), (2,1,1962), (2,2,2), (2,3,32078), (2,4,51), (2,6,11205), (2,7,354), (2,8,210), (2,20,154), (2,30,3081), (3,2,5), (3,4,4), [and others].

  The appearance of (i,j,k) in the following list means that X_i*(anticevian triangle of X_j = X_k: (1,1,40), (1,2,8), (1,3,3157), (1,6,3556), (1,8,6552), (1,9,1), (1,10,10), (1,11,21132), (2,1,165), (2,2,2), (2,3,3167), (2,6,154), (2,7,32079), [and others].

• X₄₂₄₆₄–X₄₂₄₇₁ — Perspectors associated with product triangles

  contributed by Clark Kimberling and Peter Moses, March 28, 2021. Let T₁ = A₁B₁C₁ be a triangle, and let M₁ be the matrix whose rows are the normalized barycentrics of A₁, B₁, C₁, respectively. Let T₂ = A₂B₂C₂ be a triangle, and let M₂ be the matrix whose rows are the normalized barycentrics of A₂, B₂, C₂, respectively. Let M be the matrix product M₁*M₂. The triangle product T₁*T₂ is here defined as the triangle whose vertices are given by the rows of M₁*M₂. The noncommutative operation denoted by *, here named triangle multiplication, is associative.

  Abbreviate the cevian triangle of a pont P =\(p : q : r\) as cevian(P) and the anticevian triangle of U =\(u : v : w\) as anticevian(U). Let

  T₁ = A'B'C' = anticevian(P), so that A' = \(-p : q : r\)

  T₂ = A″B″C″ = anticevian triangle(U), so that A″ = \(-u : v : w\).

  Then T₁*T₂ is perspective to ABC, and the perspector, f(P,U), is given by

  u (p u^2 - q u^2 + r u^2 - 2 r u v - p v^2 + q v^2 + r v^2 + 2 q u w + 2 p v w - p w^2 - q w^2 - r w^2) (p u^2 + q u^2 - r u^2 + 2 r u v - p v^2 - q v^2 - r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 + r w^2)

  :

  -v (p u^2 - q u^2 + r u^2 - 2 r u v - p v^2 + q v^2 + r v^2 + 2 q u w + 2 p v w - p w^2 - q w^2 - r w^2) (p u^2 + q u^2 + r u^2 - 2 r u v - p v^2 - q v^2 + r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 - r w^2)

  :

  -w (p u^2 + q u^2 + r u^2 - 2 r u v - p v^2 - q v^2 + r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 - r w^2) (p u^2 + q u^2 - r u^2 + 2 r u v - p v^2 - q v^2 - r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 + r w^2)

  The appearance of (i,j,k) in the following list means that f(X_i,X_j) = X_k: (1,1,84), (1,2,7), (1,3,1069) [and others].

  The locus of a point X =\(x : y : z\) such that f(P,X) lies on this line: at infinity is the bicevian conic of the points X₂ =\(1 : 1 : 1\) and -p + q + r : p - q + r : p + q - r, given by

  (p - q + r)(p + q - r) x^2 + 2 p (p - q - r) y z + (cyclic) = 0.

  This conic, denoted by BC(P), has center q + r : r + p : p+ q and perspector

  (p - q - r)/(p^2 - q r - r p - p q) : (q - r - p)/(q^2 - r p - p q - q r) : (r - p - q)/(r^2 - p q - q r - r p)

  If P lies inside the medial triangle, then BC(P) is an ellipse. If P lies on the nine-point circle, then BC(P) is a right hyperbola.

  The appearance of (i, [name]) in the following list means that BC(X_i) is the named conic:

  (2, Steiner inellipse), (3, nine-point circle), (5, bicevian conic of X₂ and X₃, (10, circumellipse of the medial and incentral triangles [see X₃₄₅₈₅], (115, Kiepert circumhyperbola of the medial triangle) (125, Jerabek circumhyperbola of the medial triangle)

  The appearance of (i, {\(n_{1}\), \(n_{2}\), …, \(n_{k}\)}) in the next list means that BC(X_i) is the conic that passes through the points: \(X(n_{1})\), \(X(n_{2})\), …, \(X(n_{k})\): (1, {11, 1145, 1146, 2968, 3756, 4904, 6739, 6741, 7358, 8286, 16613, 38992, 39004, 39050, 40608, 40609}); (4, {122, 3184, 13611, 39020, 40616}), [and others].

  Inverse triangles are introduced in the preamble just before X₄₂₀₀₅. If T₁ and T₂ are triangles and T₁ is non-degenerate (so that its matrix is invertible), then there exists a unique triangle T such that T₁*T = T₂, and the solution is T = inverse(T₁)*T₂. Likewise, there exists a unique triangle T such that T*T₁ = T₂, and the solution is T = T₂*inverse(T₁). Following are eight examples [omitted here].

  1. Let P =\(p : q : r\) and U =\(u : v : w\). Let T₁ = cevian(P) and T₂ = cevian(U). The triangle T = A'B'C' such that T₁*T = T₂ is given by

  A' = u (v + w) (2 p u + q u + r u + p v + r v + p w + q w) : v (u + w) (-q u - r u + p v - r v + p w + q w), (u + v) w (-q u - r u + p v + r v + p w - q w)

  B' = u (v + w) (q u - r u - p v - r v + p w + q w) : v (u + w) (q u + r u + p v + 2 q v + r v + p w + q w) : (u + v) w (q u + r u - p v - r v - p w + q w)

  C' = u (v + w) (-q u + r u + p v + r v - p w - q w) : -v (u + w) (-q u - r u + p v - r v + p w + q w) : (u + v) w (q u + r u + p v + r v + p w + q w + 2 r w)

  The locus of a point U = X =\(x : y : z\) such that T is perspective to the medial triangle is the cubic given by

  (p + r) (q + r) y^2 z + (p + q) (q + r) y z^2 + (cyclic) + 2 (p^2 + q^2 + r^2 + 3(q r + r p + p q)) x y z = 0.

  For P = X₁₄₈, this cubic is K185. For P = X₁₅₀, the cubic passes through X_i for i = 7, 8, 80, 320, 42482.

  2. Let P =\(p : q : r\) and U =\(u : v : w\). Let T₁ = cevian(P) and T₂ = cevian(U). The triangle T = A'B'C' such that T*T₁ = T₂ is given by

  A' = (q + r) (r v + q w) : (p + r) (-r v + q w) : -(p + q) (-r v + q w)

  B' = (q + r) (-r u + p w) : (p + r) (r u + p w) : -(p + q) (-r u + p w)

  C' = (q + r) (-q u + p v), -(p + r) (-q u + p v), (p + q) (q u + p v)

  The locus of a point U = X =\(x : y : z\) such that T is perspective to the medial triangle is the cubic given by

  p^2 (r (p+r) (q+r) y^2 z - q (p+q) (q+r) y z^2) + (cyclic) = 0.

  This cubic passes through X₂ for every point P. For P = X₄, the cubic is K621. The appearance of (i, {\(n_{1}\), \(n_{2}\), …, \(n_{k}\)}) in the next list means that the cubic passes through the points: \(X(n_{1})\), \(X(n_{2})\), …, \(X(n_{k})\): (1, {1,2,6,37,81,3293,17147,39949,39964})

  (3, {2,3,97,216,577})

  (4, [K621],{2,4,6,24,393,847,2052,6515})

  (5,{2,5,233,31610,36412})

  [Further examples are omitted here.]

  The triangle sum, T₁ + T₂ of two arbitrary triangles T₁ and T₂ is defined in the preamble just before X₄₂₂₈₅. The two distributive laws hold:; i.e., if T is a triangle, then

  T*(T₁ + T₂) = T*T₁ + T*T₂)

  (T₁ + T₂)*T = (T₁ + T*T₂)*T

  For arbitrary points P and U, the product triangle cevian(P)*anticevian(U) is perspective to anticevian(U)*cevian(P), and the perspector is the P-Ceva conjugate of U.

  The product A'B'C' = cevian(P)*cevian(U) is given by

  A' = (q u + r u + q v + r w) : r v (u + w) : q w (u + v)

  B' = u (q u + r u + q v + r w) : r v (u + w) : q w (u + v)

  C' = q u (v + w) : p v (u + w) : w (p u + q v + p w + q w)
  In particular, the square of cevian(P) is given by

  A' = p (p q + q^2 + p r + r^2) : q r (p + r) : q r (p + q) :

  B' = p r (q + r) : q (p^2 + p q + q r + r^2) : p r (p + q) :

  C' = p q (q + r) : p q (p + r) : r (p^2 + q^2 + p r + q r)

• X₄₂₄₇₂–X₄₂₄₈₃ — Gibert points on the cubic K1200

  contributed by Peter Moses, April 2, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉. See K1200.

• X₄₂₄₈₈–X₄₂₅₃₆ — Gibert points on the Brocard-Kiepert quartic, Q073

  contributed by Peter Moses, April 6, 2021. Gibert points are introduced in the preamble just before X₄₂₀₈₅; for the Brocard-Kiepert quartic, see Q073.

• X₄₂₅₃₇–X₄₂₅₄₆ — Gibert points on the Brocard-Kiepert quartic, Q073

  contributed by Peter Moses, April 6, 2021. Gibert points are introduced in the preamble just before X₄₂₀₈₅; for the Brocard-Kiepert quartic, see Q073.

• X₄₂₅₅₇–X₄₂₅₆₁ — Gibert points on the KHO quartic Q168

  contributed by Peter Moses, April 7, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉. This section gives four new points on the following KHO quartic (using KHO coordinates (x,y,z), introduced at KHO curves.

  2*x^4 + 30*x^2*y^2 - 108*y^4 - 27*x^2*y*z + 216*y^3*z - 171*y^2*z^2 + 81*y*z^3 - 18*z^4 = 0.

  This quartic passes through X_i for these i: 2,5,17,18,371,372,1131,1132,3068,3069,3070,3071,8976,13951,31412,39641,39642, 42557, 42558, 42559, 42560.

• X₄₂₅₆₂–X₄₂₅₉₃ — Gibert points

  contributed by Peter Moses, April 7, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉. Points X₄₂₅₆₂-X₄₂₅₇₅ lie on the cubic K458. Points X₄₂₅₇₆-X₄₂₅₇₉ lie on the cubic K369.

• X₄₂₅₉₄–X₄₂₆₀₉ — Gibert points on the cubic K1204

  contributed by Peter Moses, April 13, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉. See K1204.

• X₄₂₆₁₄–X₄₂₆₂₁ — Central angle points

  contributed by Clark Kimberling and Peter Moses, April 15, 2021. Suppose the P is a point in the plane of a triangle ABC. Let A' = angle BPC, B' = angle CPA, C' = angle APB, and assume that A' = t*A + u*(B + C) for some real numbers t and u. Since B + C = π - A, the angle A' is given by

  A' = A'(t) = t*(3A - π)/2 + π - A, and B' and C' are determined cyclically, and the corresponding point P = P(t) is given by

  P(t)= 1/(cot(A) + cot(A - t*(3*A - π)/2)) : : ,

  so that P(t) is a major center, here named the t-central angle point. Let L denote the locus of P(t) as t traverses the real number line, and note that P(2 - t) = isogonal conjugate of P(t), so that if a point X is on L, then the isogonal conjugate of X is also on L. The appearance of (t,i) in the following list means P = X_i is the point for which A'(t) = angle BPC, B'(t) = angle CPA, C'(t) = angle APB. The list is presented in isogonal conjugate pairs:

  (-2,5964); (4,5963)

  (-1,41622); (3,42621)

  (-2/3,42619; (8/3,42620)

  (0,4); (2,3)

  (1/3,42615); (5/3,42616)

  (1/2,42618); (3/2,42614)

  (2/3,13); (4/3,15)

  (1,1); (1,1)

• X₄₂₆₂₅–X₄₂₆₄₈ — Gibert points on the cubic K1191 (Evans 5th cubic)

  contributed by Peter Moses, April 16, 2021. See K1191 and the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉.

• X₄₂₆₅₈–X₄₂₆₇₁ — Points on the Lemoine axis

  contributed by Peter Moses, April 19, 2021. Suppose that P' =\(p' : q' : r'\) is a point on a line p x + q y + r z = 0 and that u x + v y + w z = 0 is a line, L. Then the point P″ = (p/u)*p’ : (q/v)*q’ + (r/w)*r’ (p/u)*p’ : (q/v)*q’ + (r/w)*r’ lies on L For example, if P' is on the Euler line and L is the Lemoine axis, X₁₈₇X₂₃₇, then P″ is on L. Points X₄₂₆₅₈-X₄₂₆₇₁ are obtained in this manner, where, in the same order, P' = X_i for i = 20, 23, 378, 429, 447, 460, 469, 858, 860, 1113, 1114, 1981, 2074, 2409.

• X₄₂₆₇₆–X₄₂₆₈₁ — Centers on cubic K588

  contributed by César Eliud Lozada, April 19, 2021. The three internal bisectors AI, BI, CI of ABC are rotated about each corresponding vertex of ABC of a same angle θ, all outwardly or all inwardly. The six rotated bisectors define a triangle N_aN_bN_c which is perspective to ABC at a point P. The locus of P is K588. (Reference: Bernard Gibert, CTC K588)

  For a given angle θ, the perspector P(θ), here denoted by K588(θ), has barycentrics coordinates:

  P(θ) = a*sin(A/2 - θ)/sin(A/2 + θ) : : , or, equivalently,

  P(t) = a*((a+b+c)*(-a+b+c)*t-S*(1-t^2))/((a+b+c)*(-a+b+c)*t+S*(1-t^2)) : :, where t = tan(θ/2)

  Some perspectors K588(θ) are shown in the following table: [table omittd here]. Note: P(θ) and P(-θ) are isogonal conjugates.

• X₄₂₆₈₂–X₄₂₆₉₅ — Gibert (i,j,k) points on cubics K1206a and K1206b

  contributed by Peter Moses, April 19, 2021. See

  K1206

  . Gibert points are introduced in the preamble just before X₄₂₀₈₅

• X₄₂₆₉₈–X₄₂₇₂₄ — Points on the line X₂X_37

  contributed by Peter Moses, April 21, 2021. Suppose that P' =\(p' : q' : r'\) is a point on a line p x + q y + r z = 0 and that u x + v y + w z = 0 is a line, L. Then the point P″ = (p/u)*p’ : (q/v)*q’ + (r/w)*r’ (p/u)*p’ : (q/v)*q’ + (r/w)*r’ lies on L For example, if P' is on the Euler line and L is the line X₂X₃₇, then P″ is on L. Points X₄₂₆₉₈-X₄₂₇₁₅ are obtained in this manner from the Euler line, where, in the same order, P' = X_i for i = 5, 20, 24, 186, 237, 297, 378, 404, 405, 406, 407, 447, 451, 458, 461, 468, 469, 475. Points X₄₂₇₁₆-X₄₂₇₂₄ are obtained from points P' on the line at infinity, with indices in this order: 30, 511, 515, 516, 518, 524, 528, 674, 1499.

• X₄₂₇₂₅–X₄₂₇₃₀ — Gibert points on cubic K1207

  contributed by Peter Moses, April 21, 2021. See

  K1207.

  Gibert points are introduced in the preamble just before X₄₂₀₈₅

• X₄₂₇₃₁–X₄₂₇₃₈ — Centers of circles that pass through X₁₃ and X₁₄

  contributed by Clark Kimberling and Peter Moses, April 22, 2021. See X₁₃ for a list of circles that pass through X₁₃ and X₁₄, and see X₁₅ for circles through X₁₅ and X₁₆.

• X₄₂₇₄₀–X₄₂₇₄₇ — Dao-Lester and Dao-Parry circles

  contributed by César Eliud Lozada, April 22, 2021. These constructions are based on two problems in the paper “Generalizations of some famous classical Euclidean geometry theorems”, by Dao Thanh Oai & als., published in International Journal of Computer Discovered Mathematics, Vol. 1, No. 3, 2016, pp. 13-20.

• X₄₂₇₅₀–X₄₂₆₇₇₂ — Points on the Sherman line

  contributed by Peter Moses, April 23, 2021. Suppose that P' =\(p' : q' : r'\) is a point on a line p x + q y + r z = 0 and that u x + v y + w z = 0 is a line, L. Then the point P″ = (p/u)*p’ : (q/v)*q’ + (r/w)*r’ (p/u)*p’ : (q/v)*q’ + (r/w)*r’ lies on L For example, if P' is on the line at infinity and L is the Sherman line, X₃₂₅₉X₃₃₂₆, then the point P'*X₁₀₀₁₅ is on L. Points X₄₂₇₅₀-X₄₂₇₇₂ are obtained in this manner from the line at infinity, where, in the same order, P' = X_i for i = 30, 511, 512, 513, 514, 515, 516, 517, 518, 523, 524, 525, 527, 528, 536, 537, 726, 740, 758, 912, 918, 926, 971.

• X₄₂₇₇₃–X₄₂₇₈₄ — Gibert (i,j,k) points on the cubic K1208

  contributed by Peter Moses, April 24, 2021. Gibert points are introduced in the preamble just before X₄₂₀₈₅. See K1208.

• X₄₂₈₁₉–X₄₂₈₈₇ — Parallels-perspeconics

  contributed by César Eliud Lozada, April 27, 2021. Let A'B'C' be a triangle perspective to ABC with perspector P. Let A_b and A_c be the points at which the parallel line to B'C' through P cuts AC and AB, respectively, and build B_c, B_a, C_a, C_b cyclically. Then these six points lie on a conic q. Swapping the triangles, the six points A'_b, A'_c, B'_c, B'_a, C'_a C'_b, constructed similarly, also lie on another conic q’.

  Conics q and q’ are introduced here as the parallels-perspeconic of ABC to A’B’C’ and the parallels-perspeconic of A’B’C to ABC, respectively.

  Assume ABC is reference triangle, the perspector P =\(x : y : z\) (barycentrics) and Q, Q' are the centers of q and q’, respectively.

  If A'B'C' is the cevian triangle of P then Q' = P and Q = (y + z)*x : :, i.e., Q is the crosspoint of X₂ and P. If A'B'C' is the anticevian triangle of P then Q = P and Q' = x*(y + z-x): :, i.e., Q' is the X₂-CevaConjugate-of-P. If A'B'C' is the circumcevian triangle of P then:

  Q = 2*(y + z)*b^2*c^2*x^3 + ((3*a^2 + b^2 - c^2)*c^2*y^2 + 4*S^2*y*z + (3*a^2 - b^2 + c^2)*b^2*z^2)*x^2 + 2*((2*y + z)*z^2*b^2 + (y + 2*z)*y^2*c^2)*a^2*x + 2*a^4*y^2*z^2 : :

  Q' = 4*(y + z)*b^2*c^2*x^3 + x^2*(c^2*y^2 + b^2*z^2)*(3*a^2 + b^2 + c^2)-x*y*z*((a^2*(a^2-b^2-c^2)-4*b^2*c^2)*x + a^2*(a^2-b^2-3*c^2)*y + a^2*(a^2-3*b^2-c^2)*z)-2*a^4*y^2*z^2 : :

  More generally: If A'B'C' is a central triangle perspective at P to ABC then there exists an homogeneous degree-0 function ƒ(a,b,c) such that PA'=ƒ(a,b,c)*PA, PB'=ƒ(b,c,a)*PB and PC'=ƒ(c,a,b)*PC. Shortening the notation to ƒ(a,b,c)=ƒ_a, ƒ(b,c,a)=ƒ_b, ƒ(c,a,b)=ƒ_c, the centers of the conics are:

  Q = ƒ_b*ƒ_c*((2*((ƒ_b + ƒ_c)*x + (y + z)*ƒ_a))*y*z*ƒ_a + (-x^2*ƒ_b*ƒ_c + y^2*ƒ_a*ƒ_c + z^2*ƒ_a*ƒ_b)*x)*x : :

  Q' = x*(ƒ_b*ƒ_c*((x + (y + z)*ƒ_a)*x^3*ƒ_b*ƒ_c - y*z*((y + 2*z)*z*ƒ_b + (2*y + z)*y*ƒ_c + 2*y*z)*ƒ_a^2) + x*ƒ_a*ƒ_b*ƒ_c*(((y^2 + 4*y*z + z^2)*ƒ_b*ƒ_c - 2*(2*y + z)*z*ƒ_b - 2*(y + 2*z)*y*ƒ_c)*x

  - ƒ_a*(y^3*ƒ_c + z^3*ƒ_b) - ((ƒ_c + 4)*ƒ_a - 2*(ƒ_b - 2)*ƒ_c)*y^2*z - ((ƒ_b + 4)*ƒ_a - 2*(ƒ_c - 2)*ƒ_b)*y*z^2) - ((ƒ_b - 1)*y^4*ƒ_c^2 + (ƒ_c - 1)*z^4*ƒ_b^2)*ƒ_a^2) : :

  The appearance of (T, i, j) in the following partial list means that the centers of the parallels-perspeconics of triangles ABC-to-T and T-to-ABC are X_i and X_j: (ABC-X3 reflections, 182, 3098), (anti-Aquila, 1001, 42819), [and others].

• X₄₂₉₄₀–X₄₃₀₃₃ — Gibert points on the KHO cubics K1215-K1217)

  contributed by Peter Moses, May 2, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉. and KHO curves and Catalog, with access to cubics K1215-K1217

• X₄₃₀₃₄–X₄₃₀₆₈ — Points on the Gergonne line

  contributed by Peter Moses, May 3, 2021. The Gergonne line of a triangle ABC is the line X₂₄₁X₅₁₄. If X is a point on the line at infinity, then the barycentric product X₇*X is on the Gergonne line. Let g(X) denote the isogonal conjugate of X. Then the X₉-isoconjugate of g(X) lies on the circumcircle. More generally, if P =\(p:q:r\), then the P-isoconjugate of the isogonal of the line at infinity is the line b*c*p + c*a*q + a*b*r = 0, and the P-isoconjugate of the line at infinity is the circumconic a^3*q*r*y*z + b^3*r*p*x*z + c^3*p*q*x*y = 0.

• X₄₃₁₁₈–X₄₃₁₉₂ — Points associated witrh co-Brocard circles

  contributed by César Eliud Lozada, May 6, 2021. If two non-equilateral triangles :g:`T’ <T’>` and :g:`T″ <T″>` share the same Brocard axis then their Brocard points, which are symmetrical with respect to this axis, lie on a circle. This circle is named here the co-Brocard circle of T’ and T” and its center lies on the common Brocard axis. (Note: the given condition is sufficient for the circularity of Brocard points, but not necessary).

  The reference triangle ABC shares its Brocard axis with these triangles: ABC-X3 reflections, 5th anti-Brocard, anti-inner-Grebe, anti-outer-Grebe, anti-1st Kenmotu-free-vertices, anti-2nd Kenmotu-free-vertices, anti-X3-ABC reflections, Apollonius, 2nd Brocard, 5th Brocard, circumsymmedial, inner-Grebe, outer-Grebe, 1st Kenmotu-free-vertices, 2nd Kenmotu-free-vertices, Lucas inner, Lucas(-1) inner, Lucas tangents, Lucas(-1) tangents, X3-ABC reflections. Other groups of triangles having common Brocard axes are showed in the following table [table omitted here].:

  In this section, a set of new anti-triangles is introduced: {anti-1st Auriga, anti-2nd Auriga, anti-Ehrmann-mid, anti-inner-Garcia, anti-1st Kenmotu-free-vertices, anti-2nd Kenmotu-free-vertices, anti-1st Parry, anti-2nd Parry, anti-3rd tri-squares-central, anti-4th tri-squares-central, anti-X3-ABC reflections, anti-inner-Yff, anti-outer-Yff}. The anti-triangle of a triangle T (denoted anti-T) is defined as the triangle A'B'C' such that T-of-A'B'C' is the reference triangle ABC. For coordinates of all triangles cited here, see the Index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html).

• X₄₃₁₉₃–X₄₃₂₁₂ — Gibert points on the cubic K1220

  contributed by Peter Moses, May 8, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others, and See K1220.

• X₄₃₂₁₃–X₄₃₂₂₅ — Perspectors involving ther inverses of triangles Gemini 15 and 16

  contributed by Clark Kimberling and Peter Moses, May 9, 2021. The vertices of the inverse of Gemini 15 are shown here:

  a (b + c) (a + b + c) : -b (a + b - c) (a + c) : -c (a + b) (a - b + c)

  -a (a + b - c) (b + c) : b (a + c) (a + b + c) : c (a + b) (a - b - c)

  -a (a - b + c) (b + c) : -b (a + c) (-a + b + c) : c (a + b) (a + b + c)

  The vertices of the inverse of Gemini 16 are shown here:

  (b + c) (a b + a c + b c) : -(a + c) (-a b + a c + b c) : -(a + b) (a b - a c + b c)

  -(b + c) (-a b + a c + b c) : (a + c) (a b + a c + b c) : -(a + b) (a b + a c - b c)}

  -(b + c) (a b - a c + b c) : -(a + c) (a b + a c - b c) : (a + b) (a b + a c + b c)}

  The appearance of (T,i) in the following list means that the triangle T is perspective to the inverse of Gemini triangle 15 and that the perspector is X_i:

  (ABC, 65), (orthic, 1824), (incentral, 37), (extouch, 72), (extangents, 65), (2nd extouch, 72), (3rd extouch, 43213), (Ayme, 43214), (orthic of intouch, 43125), (inner Conway, 43216), (2nd anti-circumperp-tangential, 43217), (anti-Wasat, 43218), (Gemini 11, 25917), (Gemini 13, 43219), (Gemini 15, 43220), (Gemini 16, 31993), (Gemini 18, 321), (Gemini 63, 43221) The appearance of (T,i) in the following list means that the triangle T is perspective to the inverse of Gemini triangle 15 and that the perspector is X_i:

  The appearance of (T,i) in the next list means that the triangle T is perspective to the inverse of Gemini triangle 16 and that the perspector is X_i:

  (ABC, 42027), (outer Garcia, 43222), (Gemini 13, 10), (Gemini 15, 43223), (Gemini 16, 43224), (Gemini 17,42), (Gemini 114, 43225)

  The locus of a point X such that the cevian triangle of X is perspective to the inverse of triangle Gemini 15 is the cubic K033 = pK(X₃₇,X₈), which passes through X_i for i = 1,4,8,10,40,65,72,3176,5930,39130,39131l.

  The locus of a point X such that the anticevian triangle of X is perspective to the inverse of triangle Gemini 15 is the cubic pK(X₁₅₀₀,X₁₀), which passes through X_i for i = 10,37,42,65,71,210,227,1826.

  The locus of a point X such that the cevian triangle of X is perspective to the inverse of triangle Gemini 16 is the cubic pK (X₁₀,X₁₉₂), which passes through X_i for i = 37,75,192,2998,21080,39467,42027.

  The locus of a point X such that the anticevian triangle of X is perspective to the inverse of triangle Gemini 16 is the cubic pK(X₅₉₄,X₃₇), which passes through X_i for i = 10,37,321,3971,22028,42027.

• X₄₃₂₆₀–X₄₃₂₇₂ — Perpsectors of triangles ABC and inverse Gemini triangles

  contributed by Clark Kimberling and Peter Moses, May 10, 2021. For an introduction to inverse triangles, see the preamble just before X₄₂₀₀₅. The appearance of (i,j) in the following list means that the triangles ABC and the inverse of Gemini triangle are perspective, and the perspector is X_j: (1,4654), (2,3679), (3,4835), (4,27483), [and others]

• X₄₃₂₈₁–X₄₃₂₉₀ — Perpsectors of Gemini triangles and their inverses

  contributed by Clark Kimberling and Peter Moses, May 12, 2021. For an introduction to inverse triangles, see the preamble just before X₄₂₀₀₅. If T = A'B'C' is a central triangle of type 1 with A' =\(u : q : r\), then the inverse of T is a central of type 1 with A-vertex given by

  (v w - q r)(u + q + r) : q (r - w) (v + r + p) : r (q - v)( w + p + q).

  (To see that this inverse is central of type 1, divide the coordinates by (r - w) (q - v).)

  The appearance of (T,j) in the following list means that the triangle T is perspective to its inverse and that the perspector is X_j: (medial,2), (anticomplementary,2), (orthic,155), (tangential,159), (incentral, 3159), (excentral, 40), (intouch,3174), (extouch,40), (Euler,4), [and others]

  The appearance of (i,j) in the following list means that the triangle Gemini i is perspective to its inverse and that the perspector is X_j:

  (1,43281), (2,100), (4,31310), (9,43282), (13,43283), (14,43284), (15,43220), (16,43224), [and others]

  Let P =\(p : q : r\) be a triangle center. The inverse of the cevian triangle of P is the anticevian triangle of q + r : r + p : p + q, and the perspector is the P'-Ceva conjugate of P″, where P' = complement of P, and P″ = anticomplement of P. The perspector is given by

  (q + r)(q^2 r + q r^2 - r^2 p - r p^2 - p^2 q - p q^2) :

  (r + p)(r^2 p + r P^2 - p^2 q - p q^2 - q^2 r - q r^2) :

  (p + q)(p^2 q + p q^2 - q^2 r - q r^2 - r^2 p - r p^2).

  The inverse of the anticevian triangle of P is the cevian triangle of - p + q + r : p - q + r : p + q - r, and the perspector is the P'-Ceva conjugate of P″, where P' = anticomplement of P, and P″ = complement of P. The perspector is given by

  p (p^3 - q^3 - r^3 + q^2 r + q r^2 - r^2 p - r p^2 - p^2 q - p q^2 - 2 p q r) :

  q (q^3 - r^3 - p^3 + r^2 p + r p^2 - p^2 q - p q^2 - q^2 r - q r^2 - 2 p q r) :

  r (r^3 - p^3 - q^3 + p^2 q + p q^2 - q^2 r - q r^2 - r^2 p - r p^2 - 2 p q r).

  For circumcevian triangles and their inverses, see the preamble just before X₄₃₃₄₄.

• X₄₃₂₉₂–X₄₃₃₂₃ — Gibert points on the cubic K1224

  contributed by Peter Moses, May 12, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others and See K1224.

• X₄₃₃₂₄–X₄₃₃₄₃ — Gibert points on the cubic K1225

  contributed by Peter Moses, May 13, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others, and See K1225.

• X₄₃₃₄₄–X₄₃₃₆₃ — Perpsectors of circumcevian triangles and their inverses

  contributed by Clark Kimberling and Peter Moses, May 13, 2021. For an introduction to inverse triangles, see the preambles just before X₄₂₀₀₅ and X43280). The perspector of the circumcevian triangle T of a point P =\(p : q : r\) and the inverse of T lies on the circumcircle. The A-vertex of T is -a^2 q r : (b^2 r + c^2 q) q : (b^2 r + c^2 q) r. The inverse of T is given by the following A-vertex:

  a^2 q r ((a^2-b^2-c^2) q r - b^2 r^2 - c^2 q^2) :

  q (b^2 r + c^2 q)((a^2 - b^2 + c^2) p r + a^2 r^2 + c^2 p^2) :

  r (c^2 p + a^2 r)((a^2 + b^2 - c^2) p q + a^2 r^2 + b^2 p^2).

  The two triangles are perspective, and their perspector, f(P), is given by

  a^2/(a^2 (q - r) + (b^2 - c^2) p) : b^2 (b^2 (r - p) + (c^2 - a^2) q) : c^2 (c^2 (p - q) + (a^2 - b^2) r).

  Let L(P) denote the line of P and X₃. If U lies on L(P), then \(f(U) = a^{2}/(a^{2}(v - w) + (b^{2}-c^{2})u)\): :

  In particular, f(Euler line) = X₁₁₀.

  The appearance of (i,j) in the following list means that f(X_iX₃) = X_j:

  (1,100), (2,110), (6,99), (7,43344), (8,901), (9,934), (10,109), (11,6099), (12,43345), (13,10409), (14,10410), [and others].

• X₄₃₃₆₄–X₄₃₃₈₇ — Gibert points on the cubic K1226

  contributed by Peter Moses, May 13, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others, and K1226

• X₄₃₃₉₀–X₄₃₃₉₆ — Points associated with the Kiss-Moses mapping

  contributed by Peter Moses, May 15, 2021. In the plane of a triangle ABC, with circumcenter O and orthocenter H, let

  D = AH∩BO, E = BH∩CO, F = CH∩AO

  D' = AH∩CO, E' = BH∩AO, F' = CH∩BO

  D″ = midpoint(D,D'), E″ = midpoint(E,E'), F″ = midpoint(F,F')

  The points D and D' and associated triangles, circles, and quadrangles comprise the Kiss-Biró configuration, introduced in “Two Remarkable Triangles of a Triangle and Their Circumcircles”, by Sándor Nagydonbai Kiss and Bálint Biró. Elemente der Mathematik (April 2020).

  Peter Moses noted that for many choices of i, the point X_i-of-D″E″F″ = X_j-of-ABC for some j. If P =\(p : q : r\) is a point and P-of-D″E″F″ = U =\(u : v : w\), then the mapping P → U is here named the Kiss-Moses mapping, and the image of P is denoted by KM(P). Moses established that this mapping is one-to-one and gave formulas for the mapping and its inverse, denoted by MK(U):

  KM(P) = a^2*(2*a^2*b^2*c^2*(a^2 - b^2 - c^2)*p + c^2*(a^2 + b^2 - c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*q + b^2*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*r) : :

  MK(U) = a^2*((a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 + 7*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - b^6*c^2 + 3*a^4*c^4 - 3*a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - b^2*c^6)*u + (a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*v + (a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)*w) : :

  The fixed point of the two mappings is X₅₄.

  The appearance of i → j in the following list means that KM(X_i) = X_j:

  15035 → 2→ 9730

  110 → 3 → 5 → 13630

  43390 → 43391 → 14157 → 74 → 4→ 185 → 13403→ 43392 → 43393

  [and others]

  If P is on the line at infinity, then KM(P) is also on the line at infinity, and KM(KM(P)) = P. The appearance of (i,j) in the following list means that X_i is on the line at infinity and KM(X_i) = X_j:

  (30,5663), (511,542), (512,690), (513,8674), (514,2774), (515,2779), (516,2772), (517,2771), (518,2836), (519,2842), (520,9033), (521,2850), (522,2773), (523,526), (524,2854), (525,9517)

  If P is on the line at infinity, then the self-inversive restriction of the mapping KM (to the line at infinity), here denoted by IKM, is given by

  IKM(P) = a^2*(c^2*(a^2 - c^2)*q + b^2*(a^2 - b^2)*r)^2*((a^2 - b^2)*(a^2 + b^2 - c^2)*q - b^2*(b^2 - c^2)*r)*(c^2*(b^2 - c^2)*q + (a^2 - c^2)*(a^2 - b^2 + c^2)*r)*(-(c^2*q^2) + (a^2 - b^2 - c^2)*q*r - b^2*r^2) : :

• X₄₃₃₉₇–X₄₃₄₁₅ — Gibert points on the cubic K1228

  contributed by Peter Moses, May 16, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others, and K1228

• X₄₃₄₁₆–X₄₃₄₃₉ — Gibert points on the cubic K1229

  contributed by Peter Moses, May 17, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others, and K1229

• X₄₃₄₄₈–X₄₃₄₅₆ — Points on the cubic K1230

  contributed by Peter Moses, May 18, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others, and K1230

• X₄₃₄₅₇–X₄₃₅₂₆ — Gibert points associated with the KHO hyperbola {{X₂,X₄,X₁₅,X₁₆,X₃₁₆}}

  contributed by Peter Moses, May 19, 2021. For KHO curves, see the preamble just before X₄₂₅₆₁. The Gibert KHO hyperbola, given by the equation

  (b - c)^2*(b + c)^2*(a^2 - b^2 - c^2)^2*x^2 + (a^2 - c^2)*(b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*c^4)*x*y + (a - c)^2*(a + c)^2*(a^2 - b^2 + c^2)^2*y^2 - (a^2 - b^2)*(b^2 - c^2)*(a^4 - 2*b^4 - 2*a^2*c^2 + c^4)*x*z - (a^2 - b^2)*(a^2 - c^2)*(2*a^4 - b^4 + 2*b^2*c^2 - c^4)*y*z + (a - b)^2*(a + b)^2*(a^2 + b^2 - c^2)^2*z^2 = 0,

  passes through X_i for these i: ,4,15,16,316,1348,1349,2039,2040,3096,5418,5420,6560,6561,7790,7859,9993,14165,36969,36970,38227,42936,42937,43195,43196, and also the 67 points X₄₃₄₆₀-X₄₃₅₂₆.

• X₄₃₅₈₄–X₄₃₅₉₆ — Perpsectors involving KM triangles

  contributed by Clark Kimberling and Peter Moses, June 2, 2021. The Kiss-Moses mapping, denoted by KM and defined in the preamble just before X₄₃₃₉₀, is a linear transformation that maps central triangles to central triangles. The appearance of (T₁,T₂,n) in the following list means that KM(T₁) is perspective to T₂ and the perspector is X_n.

  (medial, orthocentroidal, 15043)

  (medial, 3rd Hazipolakis, 22966)

  (medial, anti-orthocentroidal, 43584)

  [and others].

• X₄₃₆₃₀–X₄₃₆₄₉ — Points on the Pythagorean conic

  contributed by Peter Moses, June 7, 2021. As introduced by Bernard Gibert in K1231, the Pythagorean conic is the set of KHO points (x,y,z) that satisfy \(z^{2}\) = \(x^{2}\) + \(y^{2}\) but not x*y*z = 0. If (x,y,z) is such a point, then so are these seven points:

  (-x,y,z), (x,-y,z), (x,y,-z), (y,x,z), (-y,x,z), (y,-x,z), (y,x,-z).

  Thus, if (u,v,w) is a Pythagorean triple of integers, then then the Gibert (u,v,w) point lies on the Pythagorean conic.

• X₄₃₇₆₉–X₄₃₈₀₂ — Gibert points on the cubic K1232

  contributed by Peter Moses, June 15-17, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉, and K1232, a crunodal KHO-cubic

• X₄₃₈₀₃–X₄₃₈₁₅ — Special KM and MK perspectors

  contributed by Peter Moses and Clark Kimberling, June 20, 2021. Let KM and MK be the Kiss-Moses mapping and its inverse, as introduced in the preabmle just before X₄₃₃₉₀. For many pairs of triangles U and V, the triangle KM(U) is perspective to MK(V) and KM(V) is perspective to MK(U). In some cases, the two perspectors are the same point. This section presents several such points. The appearance of (i,U,V) in the following list means that KM(U) is perspective to MK(V) and KM(V) is perspective to MK(U) and that in both cases, the perspector is X_i [list omitted here].

• X₄₃₈₁₆–X₄₃₈₃₉ — Special KM and MK perspectors

  :contributed by Peter Moses and Clark Kimberling, June 21, 2021. Let KM be the Kiss-Moses mapping, as introduced in the preabmle just before X₄₃₃₉₀. The appearance of (T,i) in the following list means that KM(KM(T)) is perspective to ABC, and the perspector is X_i.

  (ABC,54), (medial,43816), (anticomplementary,43817), (Euler,43818), (tangential-triangle-of-1st-circumperp,43819), (tangential triangle of 2nd circumperp,43820), (X₅-reflection of ABC,43821), (Aquila,43822), [and others]

• X₄₃₈₄₅–X₄₃₈₆₈ — Special KM and MK perspectors

  contributed by Peter Moses and Clark Kimberling, June 23, 2021. Let KM be the Kiss-Moses mapping, as introduced in the preamble just before X₄₃₃₉₀. The appearance of (T,i) in the following list means that KM(KM(ABC)) is perspective to T, and the perspector is X_i: (ABC, 54), (medial, 43817), (anticomplementary, 43816), (Euler, 43831), {O-reflection of ABC, 43601), (ABC-reflection of O in ABC, 43845), (Kosnita, 54), (X₅-reflection of ABC, 43821), [and others]

  For triangles T such that ABC is perspective to K(K(T)), see the preamble just before X₄₃₈₁₆.

• X₄₃₈₆₉–X₄₃₈₉₀ — Gibert points on the cubic K1234

  contributed by Peter Moses, June 23, 2021. See also the preambles just before X₄₂₀₈₅, X₄₂₄₁₃, and X₄₂₄₂₉ and others, and K1234.

• X₄₃₉₂₁–X₄₃₉₃₃ — Points on the circumparabola with center X₅₁₃

  contributed by Peter Moses, July 2, 2021. Suppose that U =\(u : v : w\) is a point in the plane of a triangle ABC. The locus of of a point X =\(x : y : z\) such that the circumconic {{A, B, C,U,X}} is a parabola is given by the quartic equation f(u,v,w,x,y,z) + f(v,w,u,y,z,x) + \(f(w,u,v,z,x,y) = 0\), where

  \(f(u,v,w,x,y,z) = u^{2} (v + w)^{2} y^{2} z^{2} - 2 v w (-v w + u(u + v + w)x^{2} y z)\).

  If U lies on the line at infinity, then U = (b - c)(1 + b t + c t) : (c - a)(1 + c t + a t) : (a - b)(1 + a t + b t)

  for some function t that is symmetric in a,b,c and has degree 0 of homogeneity. In this case,

  \(f(x,y,z,u,v,w) = ((b - c)^{2} (1 + b t + c t)^{2} y z)^{2}\).

  The appearance of i, {\(j_{1}\), \(j_{2}\),…, \(j_{k}\)} in the following list means that the circumparabola with center X_i passes through the points \(X(j_{1})\),\(X(j_{2})\),…, \(X(j_{k})\):

  30, {30,476,3233,4240,9141,16077}

  511, {511,805,877,14966,15631}

  512, {512,669,805,875,881,886,15630,32729,38241}

  513, {513,649,660,889,901,3572,3733,4581,7192,15635,17929,17940,23345,23836,32735,35365,38242,42921—43933}

  514, {514,693,927,3676,4444,4555,4583,4608,4817,6548,6549,7192,15634,17925,17930,37143}

  516, {516,927,3234,23973,41321}

  517, {517,901,15632,23981,38243}

  518, {518,660,677,883,2284,6078}

  519, {519,4555,6079,17780,38244}

  [and others]

  {{A, B, C,523,476}} is the “X-parabola of ABC″, as in X₁₂₀₆₅.

  See Michel Bataille, Forum Geometricorum 11 (2011) 57-63, On the Foci of Circumparabolas and Bernard Gibert, Q077

  If PP is one of these circumparabolas, let P' be its perspector. Then if P″ is a point on the circumcircle, then the point P″*P'/X₆ is on PP. Various choices of P″ yeild the points X₄₂₉₂₁-X₄₃₉₃₃.

• X₄₃₉₃₅–X₄₃₉₄₅ — Points associated with circumparabola with infinite center

  ontributed by Peter Moses, July 5, 2021. In the following table [only partially shown here] each number k represents the triangle center X_k: center focus vertex perspector 30 38246 43941 3163 511 43935 43942 11672 512 38017 1084 513 38018 1015 514 38019 43943 1086 516 23972

  Barycentrics for the vertex of the circumparabola with center U =\(u : v : w\) are given by

  (v + w)*(2*c^2*v^3 - 3*(a^2 - b^2 - c^2)*v^2*w - (a^2 - 5*b^2 - c^2)*v*w^2 + 2*b^2*w^3)*(2*c^2*v^3 - (a^2 - b^2 - 5*c^2)*v^2*w - 3*(a^2 - b^2 - c^2)*v*w^2 + 2*b^2*w^3) : : ,

  this being the combo U'-(cross conjugate of U), where U' = orthocpoint of U.

  See Bernard Gibert, Q077 and Q079. See also the preamble just before X₄₂₉₂₁.

• X₄₃₉₇₀–X₄₄₀₁₀ — Points associated with Vijay parallel transforms

  contributed by Clark Kimberling (July 18, 2021), based on notes from Dasari Naga Vijay Krishna, July 13, 2021.

  Let U =\(u : v : w\) be a point in the plane of a triangle ABC. Let

  I_a = line through U parallel to line BC, and define I_b and I_c cyclically;

  A_b = I_c∩BC, and define B_c and C_a cyclically;

  A_c = I_c∩BC, and define B_a and C_b cyclically;

  U_a = reflection of U in line BC, and define U_b and U_c cyclically;

  P_a = line through U_a parallel to the line BC, and define pb and pc cyclically,

  A'B'C' = anticomplementary triangle of ABC;

  A'_b=A_bB_a∩C'A', and define B'_c and C'_a cyclically;

  A'_c=A_cA_a∩B'A', and define B'_a and C'_b cyclically.

  Define 9 points based on the preceding constructions as follows:

  A₁ = B_cC_a∩C_bB_a , B₁ = C_aA_b∩A_cC_b, C₁ = A_bB_c∩B_aA_c

  A₂ = A_bB_c∩A_cC_b, B₂ = B_cC_a∩B_aA_c, C₂ = C_aA_b∩C_bB_a

  A₃ = A_bC_b∩A_cB_c, B₃ = A_cB_c∩B_aC_a, C₃ = B_aC_a∩A_bC_b

  A₄ = B'_cB'_a∩C'_aC'_b, B₄ = C'_aC'_b∩A'_bA'_c, C₄ = A'_bA'_c∩B'_cB'_a

  A₅ = BA'_c∩CA'_b, B₅ = CB'_a∩AB'_c, C₅ = AC'_b∩BC'_a

  A₆ = A_bA'_c∩A_cA'_b, B₆ = B_cB'_a∩B_aB'_c, C₆ = C_aC'_b∩C_bC'_a

  A₇ = BC₄∩CB₄, B₇ = AC₄∩CA₄ , C₇ = AB₄∩BA₄

  A₈ = A_bC₄∩A_cB₄, B₈ = B_cA₄∩B_aC₄ , C₈ = C_aB₄∩C_bA₄

  A₉ = P_b∩P_c , B₉ = P_c∩P_a , C₉ = P_a∩P_b.

  Barycentrics for the 9 points and others [omitted here]:

  Related triangles are named as follows:

  A₁B₁C₁ = 1st Vijay parallel transform triangle of U

  A₂B₂C₂ = 2nd Vijay parallel transform triangle of U

  A₃B₃C₃ = 3rd Vijay parallel transform triangle of U

  A₄B₄C₄= 4th Vijay parallel transform triangle of U

  A₅B₅C₅ = 5th Vijay parallel transform triangle of U

  A₆B₆C₆ = 6th Vijay parallel transform triangle of U

  A₇B₇C₇ = 7th Vijay parallel transform triangle of U

  A₈B₈C₈ = 8th Vijay parallel transform triangle of U

  A₉B₉C₉ = 9th Vijay parallel transform triangle of U

  Collinearities:

  U, A, A₃, A₅, A₉ are collinear.

  U, A₁, A₂ are collinear.

  A₃, A₄, A₈ are collinear.

  Perspectors: [list omitted here]

  Barycentrics for Vijay parallel transforms:

  Vijay 1st parallel transform triangle of U

  = 1/ (u*(u + v + w) - v*w) : :

  = perspector of ABC and 4th Vijay parallel transform triangle of U
  Vijay 2nd parallel transform of U:

  (u*(u + v + w) - v*w)*((u + v + w)^2 + u^2) : :

  = perspector of ABC and 7th Vijay parallel transform triangle of U<br. [and others]
  Constructions as downloadable pdfs: Vijay 1st parallel transform (Vijay%201st%20parallel%20transform.pdf)

  [and others]

  The appearance of {i,j} in the following list means that the 1st Vijay parallel transform of X_i is X_j:

  {1,596}, {2,2}, {3,6662}, {4,68}, {6,6664}, {7,6601}, {8,4}, {20,3346}, {69,66}, {75,13476}, {76,27375}, {99,36955}, {144,42483}, {145,6553}, {192,2998}, {193,6339}, {194,42486}, {264,42487}, {329,34546}, {3869,42485}, {4329,42484}, {4560,15412}, {5905,6504}, {6360,34287}, {7057,8}, {16017,189}, {16018,39694}, {20346,7357}, {20534,7}, {40383,330}

  Also, if X is on the line at infinity, then the 1st Vijay parallel transform of X is X.

  [Lists for other Vijay parallet transforms (numbered 2 to 13) are omitted here.]

  The Vijay 1st parallel transform of U is the isogonal conjugate of the Vu tangential transform of U, and the perspector of ABC and the reflection of the cevian triangle of U in the centroid of ABCU. It is also the anticomplement of the centroid of {{[complement of U], [vertices of anticevain triangle of complement of U]}}, and also the isogonal conjugate of the TCC-perspector of [isogonal conjugate of complement of U]. (Randy Hutson, August 24, 2021)

Part 23

• X₄₄₀₂₁–X₄₄₀₂₅ — Points associated with Vijay-Hutson triangles

  contributed by Clark Kimberling (July 21, 2021), based on notes from Dasari Naga Vijay Krishna, July 21, 2021 and later from Peter Moses and César Lozada. In the plane of a triangle ABC, let

  P_a = parabola with A as its focus and BC as its directrix, define P_b and P_c cyclically;

  A' = apex of equilateral triangle erected externally on the side BC, define B' and C' cyclically;

  A″ = apex of equilateral triangle erected internally on the side BC, define B″ and C″ cyclically;

  A_b = (triangle BCA')∩P_b, and define B_c and C_a cyclically; see Vijay 3 parabolas and 6 points.png (Vijay%203%20parabolas%20and%206%20points.png) A_c = (triangle BCA') ∩P_c, and define B_a and C_b cyclically;

  A'_b = (triangle BCA″)∩P_b, and define B'_c and C'_a cyclically;

  A'_c = (triangle BCA″)∩P_c, and define B'_a and C'_b cyclically;

  A₁ = C_aC_b∩B_cB_a, B₁ = C_aC_b∩A_bA_c, C₁ = A_bA_c∩B_cB_a;

  A₂ = A_bC_b∩B_cA_c, B₂ = C_aB_a∩B_cA_c, C₂ = C_aB_a∩A_bC_b;

  A₃ = C'_aC'_b∩B'_cB'_a, B₃ = C'_aC'_b∩A'_bA'_c, C₃ = A'_bA'_c∩B'_cB'_a;

  A₄ = A'_bC'_b∩B'_cA'_c, B₄ = C'_aB'_a∩B'_cA'_c, C₄ = C'_aB'_a∩A'_bC'_b;

  Barycentrics for points defined above: [omitted here]

  Related triangles are here named as follows:

  A₁B₁C₁ = 1st Vijay-Hutson triangle;

  A₂B₂C₂ = 2nd Vijay-Hutson triangle;

  A₃B₃C₃ = 3rd Vijay-Hutson triangle;

  A₄B₄C₄ = 4th Vijay-Hutson triangle.

  Perspectors:

  X₄₄₀₂₁ = AA₁∩BB₁∩CC₁;

  X₄₄₀₂₂ = AA₂∩BB₂∩CC₂;

  X₄₄₀₂₃ = AA₃∩BB₃∩CC₃;

  X₄₄₀₂₄ = AA₄∩BB₄∩CC₄;

  X₄₄₀₂₅ = A₂A₄∩B₂B∩C₂C₄.

• X₄₄₀₃₉–X₄₄₀₄₁ — Points associated with Vijay incentral circles and excentral circles

  contributed by Clark Kimberling (July 26, 2021), based on notes from Dasari Naga Vijay Krishna, July 25, 2021. In the plane of a triangle ABC, let

  A'B'C' = excentral triangle

  O_a = circle with diameter BC, and define O_b and O_c cyclically;

  A_b = BB'∩O_a, and define B_c and C_a cyclically;

  A_c = CC'∩O_a, and define B_a and C_b cyclically;

  A'_b = A'C'∩O_a, and define B'_c and C'_a cyclically;

  A'_c = A'B'∩O_a, and define B'_a and C'_b cyclically;

  Oe = circle {{A'_b, A'_c, B'_c, B'_a, C'_a, C'_b}}, here named the Vijay excentral circle;

  OI_a = circle {{C'_a, B'_a, A_b, A_c, B_c, C_b}}, here named the Vijay a-incentral circle;

  OI_b = circle {{A'_b, C'_b, B_a, B_c, C_a, A_c}}, here named the Vijay b-incentral circle;

  OI_c = circle {{B'_c, A'_c, C_a, C_b, A_b, B_a}}, here named the Vijay c-incentral circle;

  P_a = polar of A wrt Oe, and define P_b and P_c cyclically;

  L'_a = polar of A wrt OI_a, and define L'_b and L'_c cyclically.

  Barycentric equations for Vijay incentral and excentral circles and their centers:

  Vijay excentral circle:

  (s - b)(s - c)x^2 + (s -c)(s - a)y^2 + (s - a)(s - b)z^2 + s(ayz + bzx + cxy) = 0, with center X₁₀ = b + c : c + a : a + b
  Vijay a-incentral circle:

  (s - b)(s - c)x^2 - (s)(s - b)y^2 - (s)(s - c)z^2 + (s - a)(- ayz + bzx + cxy) = 0, with center b + c : c - a : b - a
  Vijay b-incentral circle:

  -s(s - a)x^2 +(s - a)(s - c)y^2 - (s)(s - c)z^2 + (s - b)(ayz - bzx + cxy) = 0, with center c - b : c + a : a - b
  Vijay c-incentral circle:

  -s(s - a)x^2 - (s)(s - b)y^2 + (s - a)(s - b)z^2 + (s - c)(ayz + bzx - cxy) = 0, with center b - c : a - c : a + b

  Define 6 points by the following intersections :

  A₁ = B'_cB'_a∩C'_aC'_b, B₁ = C'_aC'_b∩ A'_bA'_c, C₁ = A'_bA'_c∩B'_cB'_a;

  A₂ = B_cB_a∩C_aC_b, B₂ = C_aC_b∩ A_bA_c, C₂ = A_bA_c∩B_cB_a;

  A₃ = C'_aA'_c∩ B'_aA'_b∩C_aA_c∩B_aA_b∩BC = midpoint of BC,

  B₃ = B'_aA'_b∩B'_cC'_b∩ B_aA_b∩ B_cC_b∩CA = midpoint of CA,

  C₃ = B'_cC'_b∩ C'_aA'_c∩ B_cC_b∩ C_aA_c∩AB = midpoint of AB;

  A₄ = center of Vijay a-incentral circle, and define B₄ and C₄ cyclically;

  A₅ = P_b∩P_c, and define B₅ and C₅ cyclically;

  A₆ = L'_b∩L'_c, and define B₆ and C₆ cyclically

  Barycentrics for points defined above:

  A_b = a : c - a : c, A_c =\(a : b : b\) - a

  A'_b = -a : a + c : c, A'_c = \(-a : b : a\) + b

  A₁ = \(-a(b + c) : S_{C} : S_{B}\)

  A₂ = 0 : s - c : s - b

  A₃ =\(0 : 1 : 1\);

  A₄ = b + c : c - a : b - a

  A₅ = 4*(s - a)^2*(s - b)*(s - c) - s^2*a^2 : s^2*a*b - 2*s*c*(s - a)*(s - b) : s^2*a*c - 2*s*b*(s - c)*(s - a)

  A₆ = ((b + c)*(b + c - 2*a)*(s - b)*(s - c)) : ((s - b)*(a*b*(s - c) + 2*c*(s - a)*(s - b))) : ((s - c)*(a*c*(s - b) + 2*b*(s - a)*(s - c)))

  Related triangles are here named:

  A₁B₁C₁ = Vijay excentral triangle;

  A₂B₂C₂ = intouch triangle;

  A₃B₃C₃ = medial triangle ;

  A₄B₄C₄ = Vijay abc-incentral triangle; Also, A₄B₄C₄ = Wasat triangle

  A₅B₅C₅ = Vijay polar excentral triangle;

  A₆B₆C₆ = Vijay polar incentral triangle.

  Collinearities and Perspectors [omitted here]:

  The Vijay excentral circle is the Spieker radical circle. The Vijay incentral circles are the extraversions of the Spieker radical circle. That is, the Vijay a-incentral circle is the radical circle of the incircle and the B- and C-excircles, and cyclically for the Vijay b- and c-incentral circles. (Randy Hutson, September 30, 2021)

  The Vijay excentral circle and the three Vijay incentral circles are Taylor circles of the excentral triangle. (Dao Thanh Oai, October 26, 2021)

  A₁B₁C₁ = Vijay excentral triangle is the 2nd extouch triangle of ABC (César Lozada, December 6, 2022)

• X₄₄₀₇₇–X₄₄₁₅₅ — Points of the form tgX or gtX, where X is on the Euler line

  contributed by Clark Kimberling and Peter Moses, August 8-11, 2021. For any triangle center P, let gP and tP denote the isogonal conjugate of P and the isotomic conjugate of P, respectively. Centers X₄₄₀₇₇-X₄₄₁₂₇ are points of the form tg(X), and X₄₄₁₂₈-X₄₄₁₅₅ of the form gt(X), where X lies on the Euler line. The appearance of (h,i,j,k,m) in the following list means that X_j is on the Euler line, and (X(h), X_i, X_j, X_k, X(m)) = (gtX, tX, X, gX, tgX): (6,2,2,6,76), (184,264,3,4,69), [and others]

  Recall that if L is a line, then gL and tL are conics, and tgL and gtL are lines. Each X on the Euler lines is given by a combo X₂ + k*(X₃, and the locus of gtX is the line

  4*(a^2 + b^2 + c^2)*(2 + 3*k)*S^2 X₆ - 3*a^2*b^2*c^2*(3 + J^2)*k*X₂₅, which, for each k, is a point on the line X₆X₂₅.

  The locus of tgX is the line

  16*S^4*X₆ - a^2*b^2*c^2*(a^2 + b^2 + c^2)*(J^2 - 9 - 12*k)*X₆₉, which, for each k, is a point on the line X₆X₆₉.

• X₄₄₁₅₉–X₄₄₁₇₃ — Isogonal conjugates and isotomic conjugates

  contributed by Clark Kimberling and Peter Moses, August 10-12, 2021. For any triangle center P, let gP and tP denote the isogonal conjugate of P and the isotomic conjugate of P, respectively. Centers X₄₄₁₅₉-X₄₄₁₇₃ are members of chains of points

  X, gX, tgX, gtgX, tgtgX, gtgtgX, …

  Following is a list of such chains, in which 0 signifies a point that is not in ETC:

  (41288, 0, 41286, 0, 41281, 44159, 41280, 40363, 1397, 3596, 56, 8, 7, 55, 6063, 2175, 41283, 9448, 41287, 0, 41289, 0, 41290)

  (9233, 40362, 1501, 1502, 32, 76, 6, 2, 2, 6, 76, 32, 1502, 1501, 40362, 9233, 40359)

  [and others]

• X₄₄₁₉₂–X₄₄₂₀₀ — Points associated with Vijay orthic polar triangle of circumcircle

  contributed by Dasari Naga Vijay Krishna, July 26, 2021.

  In the plane of a triangle ABC, O = circumcircle of triangle ABC. let

  O_a = circle with diameter BC, and define O_b and O_c cyclically;

  L_a = The perpendicular from X₃ of triangle ABC to side BC, define L_b and L_c cyclically;

  A₁, A₂ = L_a ∩O_a such that A₂ is nearer to A than A₁, and define B₁, B₂, C₁, C₂ cyclically;

  T_a, T'_a = polar of A₁, A₂ wrt O_a, define T_b, T'_b, T_c, T'_c cyclically;

  P_a, P'_a = polar of A₁, A₂ wrt O, define P_b, P'_b, P_c and P'_c cyclically;

  A₃ = T_b ∩ T_c, B₃ = T_c ∩ T_a, C₃ = T_a ∩ T_b;

  A₄ = T'_b ∩ T'_c, B₄= T'_c ∩ T'_a, C₄ = T'_a ∩ T'_b;

  A₅ = P_b ∩ P_c, B₅= P_c ∩ P_a, C₅ = P_a ∩ P_b;

  A₆ = P'_b ∩ P'_c, B₆= P'_c ∩ P'_a, C₆ = P'_a ∩ P'_b;

  Barycentrics: [omitted here]

  Related triangles are here named as follows:

  A₁B₁C₁ = 1st Vijay orthic polar triangle of circumcircle;

  A₂B₂C₂ = 2nd Vijay orthic polar trriangle of circumcircle;

  A₃B₃C₃ = 3rd Vijay orthic polar trriangle of circumcircle;

  A₄B₄C₄ = 4th Vijay orthic polar trriangle of circumcircle;

  A₅B₅C₅ = 5th Vijay orthic polar trriangle of circumcircle;

  A₆B₆C₆ = 6th Vijay orthic polar trriangle of circumcircle.
  The first four of those triangles have been introduced previously:

  A₁B₁C₁ is the outer Vecten triangle.

  A₂B₂C₂ is the inner Vecten triangle.

  A₃B₃C₃ is the 1st anti-Kenmotu-centers triangle.

  A₄B₄C₄ is the 2nd anti-Kenmotu-centers triangle.

  (Randy Hutson, January 11, 2022)

  Collinearities and Properties [omitted here]:

• X₄₄₂₀₂–X₄₄₂₀₅ — Centers of circles through X₂₄₀₀₇ and X₂₄₀₀₈

  contributed by Peter Moses, August 13, 2021. In addition to the Dao-Moses-Telv circle, {13,14,5000,5001,6104,6105,6106,6107,6108,6109,6110,6111,24007,24008}, the following four circles also pass thorugh X₂₄₀₀₇ and X₂₄₀₀₈:

  {{2,98,112,5913,10295,24007,24008}}, with center X₄₄₂₀₂

  {{4,107,111,671,5523,7426,9979,20410,24007,24008,41125}}, with center X₄₄₂₀₃

  {{51,115,132,24007,24008}, with center X₄₄₂₀₄}

  {{125,187,1560,6055,9730,10162,24007,24008}}, with center X₄₄₂₀₅

• X₄₄₂₁₀–X₄₄₂₉₀ — Midpoints on the Euler line

  contributed by Clark Kimberling and Peter Moses, August 16, 2021. If U and V are points on the Euler line, then their midpoint, given by the combo U + V, also lies on the Euler line.

• X₄₄₃₀₁–X₄₄₃₀₈ — Perspectors involving 1st and 2nd Savin triangles

  based on notes from Andrey Savin, August 19, 2021 and Peter Moses, August 20, 2021. Let A' = a : s - a : s - a = 2a : -a + b + c : -a + b + c (barycentrics) , and define B' and C' cyclically. The triangle A'B'C' is introduced here as the 1st Savin triangle.

  Let A″ = a : s + a : s + a = 2a : 3a + b + c : 3a + b + c (barycentrics), and define B″ and C″ cyclically. The triangle A″B″C″ is introduced here as the 2nd Savin triangle.

  1st Savin triangle (SavinTriangle1.pdf)

  2nd Savin triangle (SavinTriangle2.pdf)
  Let T₁ = 1st Savin triangle.

  T₁ is perspective to the following triangles, with perspector X₂: ABC, medial, anticomplementary, circum-medial, Gemini 1,2, 9-14, 20-24, 27, 28, 31-61, 65-70, 72-111.

  T₂ is perspective to the following triangles with perspector X₂: intouch, intangents, hexyl, infinite altitude, 6th mixtilinear, Hutson intouch, 3rd Conway, Garcia reflection, Gemini 8, Bevan-antipodal (see X₃₄₄₈₈).

  The appearance of (T,k) in the following list means that T₁ is perspective to T and the perspector is X_k:

  (Andromeda, 3677)

  (Jenkins, 5530)

  (2nd outer Soddy, 31582)

  (2nd inner Soddy, 31583)

  (anticevian of X₈₀₅₁, 44301)

  (pedal of X₂₄₈₅₁, 44302)
  Let T₂ = 2nd Savin triangle.

  T₂ is perspective to the following triangles, with perspector X₁: ABC, medial, anticomplementary, circum-medial, Gemini 1,2, 9-14, 20-24, 27, 28, 31-61, 65-70, 72-111.

  The appearance of (T,k) in the following list means that T₂ is perspective to T and the perspector is X_k:

  (excentral, 7308)

  (Gemini 7, 25430)

  (Soddy, 44303) (2nd Sharygin, 44304)

  (Aquila, 44305)

  (2nd extouch, 44306)

  (Gemini 15, 44307)

  (9th Vijay-Paasche-Hutson, 44308)

• X₄₄₃₁₁–X₄₄₃₁₉ — Points associated with the Moses-Soddy triangle

  based on notes from Peter Moses, August 21, 2021. Let A' = pole of Soddy line in the Soddy A-circle, and define B' and C' cyclically, so that A' = b - c : a - c : b - a

  B' = c - b : c - a : b - a

  C' = c - b : a - c : a - b

  The triangle A'B'C' is here named the Moses-Soddy triangle. This triangle is also the complement of the Yff contact triangle. The vertices A', B', C' lie on the cubic K927.

  The appearance of (T,n) in the following list means that A'B'C' is perspective to T, and the perspector is X_k: (ABC, 514), (medial, 1086), (orthic, 116), (incentral, 17761), (intouch, 11), (extouch, 4904), (McBeath, 44311), (symmedial, 44312), (Steiner, 1125), (3rd Euler, 11), (2nd Hatzipolakis, 44313), (Yff contact,2), (Garcia reflection, 3667), (Gemini 7, 7658), (Gemini 8, 3667), (anti-Ursa-minor, 44316), (Lemoine, 44317), (Ursa-major, 44318), (Ursa-minor, 44319), (24th Vijay-Paasche-Hutson, 44320)

  The Moses-Soddy triangle is also perspective to the Wasat triangle.

  X₂-of-A'B'C' = X₂₁₂₀₄

  X₃-of-A'B'C' = X₄₄₃₁₄

  X₄-of-A'B'C' = X₁

  X₅-of-A'B'C' = X₄₄₃₁₅

• X₄₄₃₂₈–X₄₄₃₄₉ — Steiner-ellipse-inverses of points on the Euler line

  contributed by Clark Kimberling and Peter Moses, August 23, 2021. Suppose that P is a point on the Euler line. Then P is given by the combo X₂ + t*x(3) for some t, and the Steiner-circumellipse-inverse of P is given by the following combo:

  (3*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6)*k - 16*S^4)*X₂ + 16*(1 + k)*S^4*X₃

  The appearance of (i,j) in the following list means that X_j = Steiner-circumellipse-inverse of X_i:

  (2,30), (3,401), (4,297), (5,40853), (20,441), (21,448), (22,15013), (23,40856), (24,44328), (25,15014), (26,44329), (27,447), (28,44330), (29,44331), (237,10684), (376,40884), (381,40885), (384,6660), (427,40889), (449,452), (458,35474), (468,40890), (472,11093), (473,11094), (858,35923), (2479, 2479), (2480,2480), (3543,44216), (4235,7473), (6655,21536), (8613,15781), (1113,44332), (1114,44333), (14953,37045), (37174,44228), (37188,44252)

  Continuing with a point P on the Euler line, with combo X₂ + t*x(3) for some t, the Steiner-inellipse-inverse of P is given by the following combo:

  (3*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6)*k - 16*S^4)*X₂ + 16*(1 + k)*S^4*X₃

  The appearance of (i,j) in the following list means that X_j = Steiner-inellipse-inverse of X_i: (2,30), (3,441), (4,44334), (5,297), (20,44335), (21,44336), [and others].

• X₄₄₃₅₀–X₄₄₃₆₀ — Steiner-ellipse-inverses of points on the line X₁X_3

  contributed by Peter Moses, August 24, 2021. Let \(f(a,b,c,x,y,z) = b c (b - c) (b + c - a) (x^{2} - y z)\). The inverse of the line X₁X₃ in the Steiner circumellipse is the ellipse given by

  f(a,b,c,x,y,z) + f(b,c,a,y,z x) + \(f(c,a,b,z,x,y) = 0\).

  Let \(g(a,b,c,x,y,z) = (b - c) (a^{3} - a b^{2} - a c^{2} - 2 a b c + 2 b^{2} c + 2 b c^{2}) (x^{2} - y z)\). The inverse of the line X₁X₃ in the Steiner inellipse is the ellipse given by

  g(a,b,c,x,y,z) + g(b,c,a,y,z x) + \(g(c,a,b,z,x,y) = 0\).

  The appearance of (j,.k) in the following list means that X_j lies on the line X₁X₃ and the X_k = Steiner-circumellipse-inverse of X_j:

  (1,239), (3,401), (55,40861), (57,40862), (65, 44350), (241,44351), (517,2), (940,44352), (982,44353), (1214,44354), (5662,17496)

  The appearance of (j,.k) in the following list means that X_j lies on the line X₁X₃ and the X_k = Steiner-inellipse-inverse of X_j:

  (1,3008), (3,441), (55,44355), (57,44356), (241,44357), (942, 44358), (982,44359), (1214,44360), (517,2), (5662,905)

• X₄₄₄₆₉–X₄₄₅₁₄ — Lozada-Lemoine circles

  contributed by César Eliud Lozada, based on a construction by Anton Zakharov in Mathoverflow. In the three following constructions, let K be the symmedian point X₆-of-ABC and T' = A'B'C' a triangle.

  Let A_b = B'C' ∩ AC and A_c = B'C' ∩ AB, and define B_c, C_a, B_a, C_b cyclically.

  The circle {{K, A_b, A_c}} cuts again AC and AB in A'_b and A'_c, respectively, and points B'_c, C'_a, B'_a, C'_b are defined similarly. For some triangles T' these last six points lie on a circle here named Lozada-Lemoine-circles-1A of T’.

  The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X_n: [list omitted herej]. The circle {{K, C_b, B_c}} cuts again AC and AB in A'_b and A'_c, respectively, and points B'_c, C'_a, B'_a, C'_b are defined similarly. For some triangles T' these last six points lie on a circle here named Lozada-Lemoine-circles-1B of T’.

  The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X_n: [list omitted here]. Now, let A_b, A_c be the orthogonal projections of A' in AC and AB, respectively, and define B_c, C_a, B_a, C_b cyclically.

  For certain triangles T', points A'_b, A'_c, B'_a, B'_c, C'_a, C'_b defined similarly as in (1a), lie on a circle, here named Lozada-Lemoine-circles-2A of T’.

  The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X_n:

  (ABC-X3 reflections, 576), (midheight, 44489), (orthic, 44470), (orthocentroidal, 44490), (reflection, 44491)

  Also, points A'_b, A'_c, B'_a, B'_c, C'_a, C'_b defined similarly as in (1b), lie on a circle for some triangles T'. These circles are named here Lozada-Lemoine-circles-2B of T’.

  The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X_n:

  (midheight, 44492), (orthic, 44480), (orthocentroidal, 44493), (reflection, 44494)

  Finally, let A_b, A_c be the points at which the parallel line through A' to BC cuts AC and AB, respectively, and define B_c, C_a, B_a, C_b cyclically.

  Points built as in (1a) lie on a circle for some triangles T'. These circles are named here Lozada-Lemoine-circles-3A of T’.

  The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X_n: [list omitted here].

  In the same way, points built as in (1b) lie on a circle for some triangles T'. These circles are named here Lozada-Lemoine-circles-3B of T’.

  The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X_n: [list omitted here].

• X₄₄₅₅₀–X₄₄₅₅₅ — Centers and perspectors of ellipses [SCE, line] [SIE, line]

  contributed by Peter Moses, August 24-30, 2021. bbreviations: SCE = Steiner circumellipse; SIE = Steiner inellipse. Suppose that L is a line given by u x + v y + w z = 0 (barycentrics). The SCE-inverse of L, denoted by [SCE, L], is given by

  u (x^2 - y z) + v (y^2 - z x) + w (z^2 - x y) = 0,

  and the SIE-inverse of L, denoted by [SIE, L], is given by

  (2u + v + w) (x^2 - y z) + (u + 2v + w) (y^2 - z x) + (u + v + 2w) (z^2 - x y) = 0.

  The center of [SCE, L] is -u + 2v + 2w : : = reflection of \(u : v : w\) in X₂.

  The center of [SIE, L] is 2u + 5v + 5w : : = complement of midpoint of \(u : v : w\) and X₂.

  The perspector of [SCE, L] is (2 v^2 + u w) (u v + 2 w^2) : : .

  The perspector of [SIE, L] is (4 u^2 + 9 v^2 + 4 w^2 + 11 v w + 9 w u + 11 u v)*(4 u^2 + 4 v^2 + 9 w^2 + 11 v w + 11 w u + 9 u v) : :

  The appearance of (L, k) in the following list means that X_k is the center of the ellipse [SCE, L]:

  (Brocard axis, 36900), (orthic axis, 1992), (anti-orthic axis, 4664), (Lemoine axis, 7757), (de Longchamps axis, 599), (Gergonne line, 3241), (Fermat line, 9979), (X₁X₆, 31150), (Koiller line, 31169), (X₁X₃, 44550), (Soddy line, 44551), (van Aubel line, 44552), (X₁X₅, 44553), (Napoleon axis, 44554), (Hatzipolakis axis, 44555)

  The appearance of (L, k) in the following list means that X_k is the perspector of the ellipse [SCE, L]:

  (orthic axis, 44556), (Lemoine aixs, 44557), (de Longchamps axis, 44558), (Gergonne line, 44559)

  The appearance of (L, k) in the following list means that X_k is the center of the ellipse [SIE, L]:

  (orthic axis, 597), (anti-orthic axis, 4755), (de Logchamps axis, 20582), (Gergonne line, 551), (brocard axis, 44560), (X₁X₃, 44561), (Lemoinne axis, 44562), (Soddy line, 44563), (Fermat line, 44564), (van Aubel line, 44565), (X₁X₅, 44566), (X₁X₆, 44567), (Napoleon axis, 44568), (Hatzipolakis axis, 44569), (Koiller line, 44570)

  The appearance of (L, k) in the following list means that X_k is the perspector of the ellipse [SIE, L]:

  (orthic axis, 44571), (Gergonne line, 44572)

• X₄₄₅₇₅–X₄₄₆₅₃ — and X₄₄₆₄₉-X₄₄₆₅₃, E-inverses of points on the Euler line, where E is a permutation ellipse

  contributed by Clark Kimberling and Peter Moses, September 1, 2021. Three specific permutation ellipses are described in the preamble just before X₄₁₁₃₃, as follows:

  1. The trisector ellipse passes through the points \(0:1:2\),\(0:2:1\),\(1:0:2\),\(2:0:1\),\(1:2:0\),\(2:1:0\). These six points trisect the sides of ABC. The ellipse is given by 2(x^2 + y^2 + z^2) - 5(y z + z x + x y) = 0.

  2. The unique self-dual permutation ellipse is given by x^2 + y^2 + z^2 - 4 (y z + z x + x y) = 0.

  3. The Steiner midway ellipse (SME), is, loosely speaking, the ellipse midway between the Steiner inellipse (SIE) and the Steiner circumellipse (SCE). That is, for each P on SCE, let P' be the intersection of the ray GP with SIE, and let P″ be the midpoint of PP'. Then SME is the set of all such midpoints. SME is given by 7(x^2 + y^2 + z^2) - 34(y z + z x + x y) = 0.

  In general, the inverse of a point P =\(p : q : r\) in an ellipse j (x^2 + y^2 + z^2) - k (y z + z x + x y) = 0 is the point k*p^2 - j*p*q + j*q^2 - j*p*r - k*q*r + j*r^2 : : .

  The center of every permutation ellipse E is X₂, so that the E-inverse of each point P on the Euler line is also on the Euler line. If P = X₂ + t*X₃ and E is give by h*(x^2 + y^2 + z^2) - k (y z + z x + x y) = 0, then

  (E-inverse of P) = (4 (h - k) S^4 + 3 t ((2 h - k) S^4 - (2 h + k) SA SB SC SW)) X₂ - 4 (h - k) (1 + t) S^4 X₃.

  The appearance of {i,j} in the following list means that X_j = trisector-ellipse-inverse of X_i:

  {3,44575}, {4,44576}, {5,44577}, {297,3545}, {376,44578}, {381,44579}, {401,5054}, {441,10304}, {448,15671}, {3524,40884}, {3839,44216}, {5055,40885}, {15699,40853}, {37907,40856}

  The appearance of {i,j} in the following list means that X_j = Steiner-midway-ellipse-inverse of X_i:

  {297,3860}, {401,44580}, {441,15690}, {12101,44216}, {15685,44335}, {15759,40884}, {19710,44346}, {33699,44334}

  The appearance of {i,j} in the following list means that X_j = self-dual-ellipse-inverse of X_i: [list omitted here].

  The appearance of {i,j} in the following list means that X_j = Steiner-circumellipse-inverse of X_i: [list omitted here].

  The appearance of {i,j} in the following list means that X_j = Steiner-incircle-ellipse-inverse of X_i: [list omitted here].

• X₄₄₅₈₂ — Kenmotu-centers triangles

  contributed by César Eliud Lozada, September 1, 2021. Let A', B', C' be the centers of the inner-Kenmotu squares, as showed in MathWorld’s Kenmotu Point. Triangle A'B'C' is named here the 1st Kenmotu-centers triangle of ABC. As there exists an outer version of these squares with centers A″, B″, C″, the triangle A″B″C″ is refered here as the 2nd Kenmotu-centers triangle of ABC.

  Barycentric coordinates of A' and A″ are:

  A' = a^2+2*S : b^2 : c^2

  A″ = a^2-2*S : b^2 : c^2

  Both triangles are homothetic to ABC with homothetic center X₆. A'B'C' is the pedal triangle of X₃₇₁-of-ABC with respect to the 1st Kenmotu-diagonals triangle and A″B″C″ is the pedal triangle of X₃₇₂-of-ABC with respect to the 2nd Kenmotu-diagonals triangle.

  In the following list, (T, i, j) means that pairs of triangles (T, A'B'C'), (T, A″B″C″) are perspective with perspectors X_i and X_j, respectively: [list omitted here]

• X₄₄₆₈₇–X₄₄₇₂₉ — Dao-conjugates

  contributed by César Eliud Lozada, based on notes from Dao Thanh Oai, September 8, 2021. Let :g:`ABC << A B C >>` be a triangle, P a point on its plane and and Ω an arbitrary circumconic of :g:`ABC << A B C >>`. Lines :g:`AP <[ A P ]>`, :g:`BP <[ B P ]>`, :g:`CP <[ C P ]>` cut again Ω at :g:`A’ <A’>`, :g:`B’ <B’>`, :g:`C’ <C’>`, respectively, and parallel lines through these points to :g:`BC <[ B C ]>`, :g:`CA <[ C A ]>`, :g:`AB <[ A B ]>` cut Ω again at :g:`A″ <A″>`, :g:`B″ <B″>`, :g:`C″ <C″>`, respectively. Then lines :g:`AA″ <[ A A″ ]>`, :g:`BB″ <[ B B″ ]>`, :g:`CC″ <[ C C″ ]>` concur. (Dao Thanh Oai, May 12, 2021).

  If barycentric coordinates of the center X of Ω are X =\(x : y : z\) and P =\(p: q : r\), then D, the point of intersection of AA″, BB″, CC″, is:

  D = D(X, P) = x*(x - y - z)*q*r : :

  For a given X, this transformation is an involution. D(X, P) is introduced here as the X-Dao conjugate of P. More precisely, D(X, P) may be expressed as an :term:`isoconjugate <isoconjugate>`:

  (X-Dao conjugate of W) = W-isoconjugate of P, where W = W(X)= a^3*(x - y + z)*(x + y - z)*y*z : :

  The appearance of (m, n) in the following list means that (X(m)-Dao conjugate of P) = X_n-isoconjugate of-P for all points P: (1, 56), (2, 31), (3, 1), (4, 19614), (5, 2190), (6, 19), (8, 16945), (9, 6), (10, 58), (11, 109), (31, 7357), (37, 1333), (57, 2192), (66, 19616), (113, 36053), (114, 36051), (115, 163), (119, 36052), (124, 36050), (125, 162), [and others].

  The X-Dao conjugate of P is the trilinear quotient P / W, and, among all known kinds of conjugaties, the X-Dao conjugate of P is: the isogonal conjugate of P, when Ω is the circumcircle of ABC (with center X = X₃), the isotomic conjugate of P, when Ω is the Steiner circumellipse of ABC (with center X = X₂), the polar conjugate of P, when Ω is the circumconic of ABC with center X = X₁₂₄₉. This conic is described in X1249.

  This figure (DaoThreeCubics.png) shows the locus of a point P such that for X = X₁, we have D(P,X) = D(X,P). Comparable loci result for other choices of X. In general, each such locus consists of three cubics, each of which passes through the vertices of the medial triangle, through X, and through the X₂-Ceva conjugate of X. Also, the A-cubic passes through A, the B-cubic passes through B, and the C-cubic passes through C.

• X₄₄₇₃₃–X₄₄₇₄₃ — Points associated with Vijay ellipses

  based on notes contributed by Dasari Naga Vijay Krishna, September 6, 2021. In the plane of a triangle ABC, let M_aM_bM_c = medial triangle and A″B″C″ = orthic triangle, and let E_a denote the ellipse that passes through A and has foci M_b and M_c; clearly E_a passes through M_a and A″. Define ellipses E_b and E_c cyclically. Names and barycentric equations for these ellipses follow:

  E_a = Vijay A-ellipse: (a^2 - b^2 + c^2) y^2 + (a^2 + b^2 - c^2) z^2 - 2 c (b + c) x y - 2 b (b + c) y z = 0

  E_b = Vijay B-ellipse: (b^2 - c^2 + a^2) z^2 + (b^2 + c^2 - a^2) x^2 - 2 a (c + a) y z - 2 c (c + a) z x = 0

  E_c = Vijay C-ellipse: (c^2 - a^2 + b^2) x^2 + (c^2 + a^2 - b^2) y^2 - 2 b (a + b) z x - 2 a (c + b) x y = 0

  See Vijay A-ellipse (Vijay%20A-ellipse.pdf), Vijay B-ellipse (Vijay%20B-ellipse.pdf), Vijay C-ellipse (Vijay%20C-ellipse.pdf).

  Let

  A_b = the point, other than A, in E_a∩AB, and define B_c and C_a cyclically;

  A_c = the point, other than A, in E_a∩AC, and define B_a and C_b cyclically;

  A₁ = A_bC_b∩A_cB_c, and define B₁ and C₁ cyclically;

  A₂ = B_aB_c∩C_aC_b, and define B₂ and C₂ cyclically;

  A₃ = A_bB_c∩A_cC_b, and define B₃ and C₃ cyclically;

  A₄ = A_bC_a∩A_cB_a, and define B₄ and C₄ cyclically;

  A₅ = the point in E_b∩E_c that is closer to A than the other point in E_b∩E_c, and define B₅ and C₅ cyclically;

  A₆ = the point in E_b∩E_c, other than A₅, and define B₆ and C₆ cyclically;

  A₇ = BC∩A₅A₆, and define B₇ and C₇ cyclically.

  Barycentrics for the nine points just above are as follows: [omtted herejj]. Collinearities:

  A₂, A₃, A₄ are collinear.

  A₅, A₆, A₇ are collinear.

  X₄₄₇₃₃, X₄₄₇₃₄, X₄₄₇₃₉, X₄₄₇₄₁ are collinear.

  X₄₄₇₃₃, X₄₄₇₃₅, X₄₄₇₃₈ are collinear.

  X₄₄₇₃₃, X₁₆₅₉, X₄₄₇₄₂ are collinear.

  X₄₄₇₃₃, X₁₃₃₉₀, X₄₄₇₄₃ are collinear.

  X₂, X₄, X₄₄₇₃₄, X₂₇ are collinear.

  X₁₆₅₉, X₁₃₃₉₀, X₄ are collinear.

  X₄, X₄₄₇₄₂, X₄₄₇₄₃ are collinear.

  Collinearities and Properties are omitted here.

  The triangle AnBnCn is here named the nth Vijay triangle, for n = 1,2,3,4,5,6,7.

• X₄₄₈₀₅–X₄₄₈₂₇ — Centers of circumcircle-inverses of lines

  contributed by Clark Kimberling and Peter Moses, September 17, 2021. The circumcircle-inverse of a line is a circle. (If the line, L, passes through X₃, the circle is has infinite radius; i.e. the inverse is the line L itself.) Suppose that a line L is the trilinear polar of a point \(p : q : r\). The circumcircle-inverse of L is the circle with center X and A-power given as follows:

  X = a^2*(c^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*p*q + b^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*p*r - a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*q*r) : :

  A-power: a^2*b^2*c^2*q*r / (b^2*(a^2 - b^2 + c^2)*p*r + c^2*(a^2 + b^2 - c^2)*p*q - a^2*(a^2 - b^2 - c^2)*q*r)).

  The appearance of {{i,j}, {\(n_{1}\), \(n_{2}\), …, \(n_{k}\)}, m} in the following list means that the circumcircle-inverse of the line X_iX_j passes through the points \(X(n_{1})\), \(X(n_{2})\), …, \(X(n_{k})\) and has center X(m).

  {{1,6}, {3,36,187,667,5144,11650,32625,32758},39227}

  {{1,21}, {3,36,501,1325,1326,3733,5161,5197,5867},39210}

  {{2,6}, {3,23,187,353,669,5866,5867,5937,5938,5939,5940,5941,6031,32531,34014},5926}

  {{2,11}, {3,23,100,105,659,1155,5144,9471,14664,14667,19915,19916},44805}

  {{2,98}, {3,23,98,110,2080,13558,14652,14673,19165,33900,42667,42668},14270}

  {{2,99}, {3,23,99,111,2079,2930,5104,11641,11643,14669,14678,14682,15564,18773,18774,33998},11616}

  [and others].

• X₄₄₈₄₀–X₄₄₈₅₉ — Perspectors involving 8th and 9th mixtilinear triangles

  contributed by César Eliud Lozada, September 18, 2021. Dan Reznik defines 8th and 9th mixtilinear triangles as follows:

  Let ω’_a, ω’_b and ω’_c be the A-, B- and C- mixtilinear incircles of ABC, and Ω_i the inner Apollonius circle of them. Denote as A', B', C' the touchpoints of Ω_i and ω’_a, ω’_b and ω’_c, respectively. The triangle A'B'C' is the 8th-mixtilinear triangle of ABC. Let ω”_a, ω”_b and ω”_c be the A-, B- and C- mixtilinear excircles of ABC, and Ω_o the outer Apollonius circle of them. Denote as A″, B″, C″ the touchpoints of Ω_o and ω”_a, ω”_b and ω”_c, respectively. The triangle A″B″C″ is the 9th-mixtilinear triangle of ABC.

  Barycentric coordinates of the A-vertices of these triangles are:

  8th mixtilinear triangle: A' = a*(a^2 + 2*(b + c)*a - 3*(b - c)^2)/2 : b^2*(a - b + c) : c^2*(a + b - c) 9th mixtilinear triangle: A″ = a*(a^2 - 2*(b + c)*a - 3*(b - c)^2)/2 : b^2*(a + b - c) : c^2*(a - b + c)

  The appearance of (T, i) in the following list means that triangles 8th mixtilinear and T are perspective with perspector X_i: [list omitted here].

  The appearance of (T, i) in the following list means that triangles 9th mixtilinear and T are perspective with perspector X_i: [list omitted here].

  The following pairs of triangles are orthologic: (8th mixtilinear, 4th mixtilinear) and (9th mixtilinear, 3rd mixtilinear). No known triangle was found to be parallelogic to either A'B'C' or A″B″C″. The 9th mixtilinear triangle is also directly similar to these triangles: excenters-midpoints, Garcia-reflection (named Gemini 8 too) and 2nd Schiffler.

• X₄₄₈₈₆–X₄₄₈₉₆ — Line conjugates on the Euler line

  contributed by Clark Kimberling and Peter Moses, September 25, 2021. See the preamble just before X₂₃₇ for an introduction to line conjugates. If P and U are points on the Euler line then the P-line conjugate of U is identical to the X₂-line conjugate of U, and if U = X₂ + k X₃, then the X₂-line conjugate of U is given by the combo

  3*a^2*b^2*c^2*(3*a^2*b^2*c^2*(3 + J^2)*k + 4*(a^2 + b^2 + c^2)*S^2)*X₂ - 4*S^2*(3*a^2*b^2*c^2*(a^2 + b^2 + c^2)*k + 4*(b^2*c^2 + a^2*(b^2 + c^2))*S^2)*X₃.

• X₄₄₉₁₆–X₄₄₉₃₄ — Hatzipolakis-Lozada circles

  contributed by César Eliud Lozada, October 2, 2021. Let :g:`ABC << A B C >>` be a triangle, :g:`A’B’C’ << A’ B’ C’ >>` the :term:`orthic triangle <orthic-triangle>` of :g:`ABC << A B C >>`, :g:`P <P>` a point and :g:`A″B″C″ << A″ B″ C″ >>` the circumcevian triangle of P. Then the circumcircles of :g:`AA’A″ << A A’ A″ >>`, :g:`BB’B″ << B B’ B″ >>`, :g:`CC’C″ << C C’ C″ >>` are coaxial and their radical axis is the line :ref:`X<sub>4</sub> <X<sub>4</sub>>`:g:`P <P*>`. (Antreas Hatzipolakis, euclid 2510, Sept. 27, 2021). Assume that O and H are the circumcenter and orthocenter of ABC, respectively.

  For P =\(x : y : z\), the radical trace T(P) of the three circles is:

  T(P)= SA*x*(y*SB^2+SC^2*z)+SB*SC*(y^2*SB+z^2*SC-2*SA*x*(y+z))+(SB+SC)*(SA^2*x^2-SB*SC*y*z) : : If P lies on the nine-point circle of ABC then T(P) = P. If P lies in the infinite, then T(P) lies on the circle with diameter OH. The appearance of (i, j) in the following list means that T(X_i)=X_j:

  (30, 3), (511, 31848), (512, 31850), (513, 31849), (514, 31851), (515, 31866), (516, 31852), (517, 31847), (518, 31865), (519, 31853), (523, 4), (524, 15098), (525, 43389), (542, 31854), (900, 18341), (952, 18342), (1503, 18338), (2574, 43396), (2575, 43395), (3564, 18348), (3566, 18347), (5965, 31864) he appearance of (i, j) in the following partial list means that T(X_i)=X_j, for P other than specified in (2) and (3):

  (1, 16869), (2, 5159), (3, 10257), (5, 2072), [and others]. When P moves on a line L through H, T(P) moves also on the line L. In particular, if P is a point on the Euler line of ABC and such that OP/OH = t, then OT/OH = (4*R^2-SW)/(2*OH*(1-t)). When P moves on a line L* no passing through H, the locus of T(P) is a circle Ω(L*), here named the Hatzipolakis-Lozada circle of line L*. Ω(L*) passes through H and, if P* is the trilinear pole of L*, then the center O* of Ω(L*) is the dilation of X₃₀₉₁ with center the polar-conjugate-of-P and ratio 5/4. If L* is the orthic axis X₂₃₁X₂₃₂ of ABC then O*=X₃₈₁; if L* is the Hatzipolakis axis X₅X₅₂₃ of ABC then O*=X₂₃₃₂₃. For other named or notable lines of ABC, O* are given in X₄₄₉₁₆ to X₄₄₉₃₂. The transformation L* → Ω(L*) is not an inversion in a circle.

  A generalization follows: Let t be a real number and let A'B'C' be the circumcevian triangle of a point P. Let A″ be the point given by the vector equation PA″ = t*PA', and define B″ and C″ cyclically. Denote the triangle A″B″C″ by T(P,t). Suppose that U is a point other than P and k is a real number, and let A’’’B’’’C’’’ = T(U,k). The circles (AA”A’’’), (BB’’B’’’), (CC’’C’’’) are coaxial, and their radical axis is the line PU. In the construction above, P is arbitrary, U = X₄, t = 1, and k = 1/2. (Vu Thanh Tung, October 4, 2021)

• X₄₄₉₃₇–X₄₄₉₆₆ — Hatzipolakis-Suppa circle

  contributed by Antreas Hatzipolakis and Ercole Suppa, October 4, 2021. Let P be a point in the plane of a triangle ABC, and let

  H = X₄ = orthocenter of ABC;

  A'B'C' = pedal triangle of H;

  A″B″C″ be the pedal triangle P;

  (O_a) = circumcircle of, AA'A″, and define (O_b) and (O_c) cyclically;

  \(R_{a}\) = radical axis of (O_b) and (O_c), and define \(R_{b}\) and \(R_{c}\) cyclically;

  (O'_a) = reflection of (O_a) in AA', and define (O'_b) and (O'_c) cyclically;

  R’a = radical axis of (O'_b) and (O'_c), and R’b and R’c cyclically;

  R’1 = reflection of R’a in BC, and define R’2 and R’3 cyclically.

  The locus of P such that the parallels to R’1, R’2, R’3 through A', B', C', resp. concur is the union of the Euler line and the circle(X₃₈₂, R), where R = circumradius of ABC.

  The circle (X₃₈₂, R) is here named the Hatzipolakis-Suppa Circle, or HS-circle.

  Let Q=Q(P) be the point in which the three parallels concur. The following properties hold:

  1. (X₃₈₂, R) is the reflection of the circumcircle in the orthocenter H.

  2. The appearance of {i, j} in the following list means that P=X_i lies on Euler line and Q(P)=X_j: {4,235}, {30,403}, {382,4}, {3146,5}, {3543,1596}, {5073,16868}, {10296,37984}, {10736,1312}, {10737,1313}, {12173,3089}, {15682,427}, {15684,7577}, {35480,25}, {35490,37197}, {44438,6623}

  3. The appearance of {i, j} in the following list means that P=X_i lies on HS-circle and Q(P)=X_j: {10152,133}, {10721,125}, {10722,115}, {10723,114}, {10724,119}, {10725,118}, {10726,124}, {10727,116}, {10728,11}, {10729,5511}, {10730,5510}, {10731,25640}, {10732,117}, {10733,113}, {10734,5512}, {10735,132}, {10736,1312}, {10737,1313}, {14989,3258}

  4. If P lies on HS-circle then Q(P) is the image of P under the homothety with center H and factor k = -1/2.

  5. When P moves on the Euler line the locus of point Q(P) is the Euler line.

  The function that maps the Euler line to the Euler line is here named the 1st Hatziplakis-Suppa transform (or 1st HS transform).

  6. When P moves on HS-circle the locus of Q(P) is the nine-point circle of ABC.

  The function that maps the HS-circle to the nine-point circle is here named the “2nd Hatzipolakis-Suppa” transform (or 2nd HS transform).

  7. Let L₁ = radical axis of (HS-circle, circumcircle); then

  X₁₆₀₈₀ = trilinear pole (L₁)

  X₄ = radical trace (HS-circle,circumcircle)
  8. Let L₂ = radical axis of (HS-circle, ninepoint-circle); then

  X₃₈₂₅₃ = trilinear pole of L₂

  X₁₃₄₇₃ = radical trace of (HS-circle,ninepoint-circle).

  9. Other points on HS-circle or on the nine-point circle are X₄₄₉₃₇-X₄₄₉₄₆ and X₄₄₉₆₇-X₄₄₉₉₂.

  10. Other points on the Euler line are X₄₄₉₄₇-X₄₄₉₅₆.

• X₄₅₀₁₀–X₄₅₁₂₄ — 1st and 2nd Aubert points

  contributed by César Eliud Lozada, October 6, 2021, based on a proposal by Ivan Pavlov. A complete quadrilateral Q formed by four lines ℓ₁, ℓ₂, ℓ₃, ℓ₄ is denoted here as Q = [ℓ₁, ℓ₂, ℓ₃, ℓ₄], or also as Q = [ABCD], if it is composed by lines AB, BC, CD, DA.

  In every complete quadrilateral Q there are four component triangles, each bounded by three different lines of Q. It is well known that the orthocenters of these triangles are collinear on a line called the Aubert line (or Steiner line) of Q.

  Let T ‘=A'B'C' and T″=A″B″C″ be two non-homothetic distinct triangles and let ℓ_a, ℓ_b, ℓ_c be the Aubert lines of quadrilaterals [B’C'B″C″], [C’A'C″A″] and [A’B'A″B″], respectively. Then ℓ_a, ℓ_b, ℓ_c concur in a point here named the 1st Aubert point of T’ and T”. (Note: If T' and T″ are homothetic, the described Aubert lines bound a triangle homothetic to T and T″).

  In the following list, (T, i) means that the 1st Aubert point of ABC and triangle T is X_i. [list omitted here]. Let T ‘=A'B'C' and T″=A″B″C″ be two triangles and let ℓ_a, ℓ_b, ℓ_c be the Aubert lines of quadrilaterals [C’A'B'A″], [A’B'C'B″], [B’C'A'C″], respectively. Let A* = ℓ_b∩ℓ_c, B* = ℓ_c∩ℓ_a and C* = ℓ_a∩ℓ_b. Then, for some pairs of triangles T ‘ and T″, the lines A*A', B*B' and C*C' concur in a point here introduced as the 2nd Aubert point of T’ to T”.

  In the following list, (T, i) means that the 2nd Aubert point of ABC to triangle T is X_i: [list omitted here].

  Similarly, in the below list, (T, i) indicates that the 2nd Aubert point of triangle T to ABC is X_i: [list omitted here].

• X₄₅₁₈₈–X₄₅₁₉₄ — V-perspeconics

  contributed by César Eliud Lozada, October 15, 2021. Let T' = A'B'C' and T″ = A″B″C″ be two perspective triangles, neither inscribed in the other. Denote A'_b = A'B″ ∩ B'C' and A'_c = A'C″ ∩ B'C', and cyclically B'_c, B'_a, C'_a, C'_b. Then these six points lie on a conic, here introduced as the V-perspeconic of T’ to T”. A reciprocal V-perspeconic T″ to T' is found by swapping T' and T″ in the previous construction.

  The appearance of (T, i, j) in the following list means that the centers of the V-perspeconics of ABC-to-T and T-to-ABC are X_i and X_j, respectively:

  (ABC-X3 reflections, 17807, 45188), (Ehrmann-mid, 45191, 45192), (outer-Garcia, 3588, 45189), (Gemini 107, 2, 2), (Gemini 109, 2, 2), (Gemini 110, 2, 2), (Gemini 111, 2, 2), (Johnson, 31353, 45190), (5th mixtilinear, 45193, 45194)

• X₄₅₁₉₆–X₄₅₂₁₅ — Perspectors of Electra(B) triangles and cevian triangles

  contributed by Clark Kimberling and Peter Moses, October 17, 2021. In this paragraph (but not the next), all coordinates are trilinears. Suppose that P =\(p : q : r\) and U =\(u : v : w\), where neither P nor U lies on a sideline of ABC. Let L(p,q,r,u,v,w) = p^2 * (v^2 + w^2) - u*(q v + r w), so that (as in the Glossary), the point \(L(u,v,w) : L(v,w,u) : L(w,u,v)\) is the P-line conjugate of U, which lies on this line: PU. Let A'B'C' be the anticevian triangle of U, so that A' = \(-u : w : w\), and B' and C' are determined cyclically. Let

  A″ = P-line conjugate of A', and define B″ and C″ cyclically. The triangle A″B″C″ is here named the (P,U)-Electra triangle. If P = X₁ =\(1:1:1\), then the locus of a point X =\(x : y : z\) such that the cevian triangle of X is perspective to the (X₁,U)-Electra triangle is given by

  v (u + w) (v^2 + w^2 + u v - u w) y^2 z - w (u + v) (v^2 + w^2 + u w - v w) y z^2 + (cyclic) - (v - w)(w - u)(u - v)(u + v + w) x y z = 0

  Next, suppose that all coordinates in the preceding paragraph are barycentrics. The triangle A″B″C″ is then the (X₁,U)-Electra(B) triangle. The appearance of (i,j,k)) in the following list means that the (X₁,X_i)-Electra(B) triangle is perspective to the cevian triangle of X_j, and the perspector is X_k:

  (1,2,4357), (1,69,3687), (1,85,45196), (1,86,2), (1,1909,10), (1,6384,45197), (1,28660,76)

  (3,2,45198), (3,95,2), (3,9291,5))

  (4,2,41005), (4,264,2), (4,1975,3), (4,6340,45199), (4,6527,46200)

  (6,2,6656), (6,83,2), (6,315,46201), (6,9230,141)

  [and others].

• X₄₅₂₁₆–X₄₅₂₃₆ — Perspectors of Electra(B) triangles and cevian triangles

  contributed by Clark Kimberling and Peter Moses, October 18, 2021. Let A″B″C″ be the (X₁,U)-Electra triangle, defined (in terms of trilinear coordinates) in the preamble just before X₄₅₁₉₆. In barycentric coordinates, the locus of X =\(x : y : z\) such that the cevian triangle of X is perspective to the (X₁,U)-Electra triangle is the cubic given by the following equation:

  a^3 c v (c u + a w) (a c^2 v^2 + a b^2 w^2 + b c^2 u v - b^2 c u w) y^2 z

  - a^3 b w (b u + a v) (a c^2 v^2 + a b^2 w^2 + b^2 c u w - b c^2 v w) y z^2

  + (cyclic) - a b c (c v - b w)(a w - c u)(b u - a v)(b c u + c a v + a b w) x y z = 0.

  The appearance of (i,j,k)) in the following list means that the (X₁,X_i)-Electra triangle is perspective to the cevian triangle of X_j, and the perspector is X_k:

  (2,1,2309), (2,56,39780), (2,86,1), (2,171,42), (2,171,42), (2,38832,31)

  (3,1,1858), (3,21,1), (3,1940,65)

  (6,1,3666), (6,7,41003), (6,63,960), (6,81,1), (6,314,75), (6,894,k37), (6,4296,1214)

  [and others]

• X₄₅₂₃₈–X₄₅₂₅₅ — Vu-Miquel points

  contributed by César Eliud Lozada, October 20, 2021. Let :g:`ABC << A B C >>` be a triangle and :g:`A’ <A’>`, :g:`B’ <B’>`, :g:`C’ <C’>` three any points on the sidelines :g:`BC <[ B C ]>`, :g:`CA <[ C A ]>`, :g:`AB <[ A B ]>`, respectively. Miquel theorem states that three circles {{AB'C'}}, {{BC'A'}}, {{CA'B'}} have a common point.

  Consider now two points P₁ and P₂. Then the three conics ({AB'C'P₁P₂}}, {{BC'A'P₁P₂}} and {{CA'B'P₁P₂}} have a common point other than P₁ and P₂. (Vu Thanh Tung, Euclid 2699, October 12, 2021).

  The common intersection of that three conics is named here the Vu-Miquel point of (P₁, P₂) with respect to {A', B', C'}.

  This section deals with Vu-Miquel points of (P₁, P₂) with respect to {A', B', C'} when A', B', C' are the cevian traces of a point P. Assume A'B'C' is the cevian triangle of a point P =\(U : V : W\) (trilinear coordinates used here), P₁ =\(u_{1} : v_{1} : w_{1}\) and P₂ =\(u_{2} : v_{2} : w_{2}\). Then the common point of the three conics is:

  Q(P, P₁, P₂) = ((U*\(d_{u}\) + V*\(d_{v}\) + W*\(d_{w})\)*U*V*W*\(s_{u}\) + (V^2*W^2*\(u_{1}\)*\(u_{2}\) - (V^2*\(w_{1}\)*\(w_{2}\) + W^2*\(v_{1}\)*\(v_{2})\)*U^2)*\(d_{u})\)*\(u_{1}\)*\(u_{2}\)/(V*\(d_{v}\) + W*\(d_{w})\) : :

  where \(d_{u}\), \(d_{v}\), \(d_{w}\) are the cyclic differences:

  \(d_{u}\) = \(v_{1}\)*\(w_{2}\) - \(v_{2}\)*\(w_{1}\), \(d_{v}\) = \(w_{1}\)*\(u_{2}\) - \(w_{2}\)*\(u_{1}\), \(d_{w}\) = \(u_{1}\)*\(v_{2}\) - \(u_{2}\)*\(v_{1}\)

  and \(s_{u}\), \(s_{v}\), \(s_{w}\) are the cyclic sums:

  \(s_{u}\) = \(v_{1}\)*\(w_{2}\) + \(v_{2}\)*\(w_{1}\), \(s_{v}\) = \(w_{1}\)*\(u_{2}\) + \(w_{2}\)*\(u_{1}\), \(s_{w}\) = \(u_{1}\)*\(v_{2}\) + \(u_{2}\)*\(v_{1}\)

  In the following list, (i, j, k, n) means that the Vu-Miquel point of (X_i, X_j) with respect to the cevian traces of X_k is X_n: (1, 2, 1, 192), (1, 3, 1, 3157), (1, 4, 1, 1148), (1, 5, 1, 45238), (1, 6, 1, 55), (1, 7, 1, 31526), (1, 8, 1, 19582), (1, 9, 1, 3158), (1, 10, 1, 3159), (1, 11, 1, 523), (1, 15, 1, 202), (1, 16, 1, 203), (1, 19, 1, 204), (1, 20, 1, 45239), (2, 6, 1, 45240), [and others]

• X₄₅₂₅₆–X₄₅₂₆₅ — 1st and 2nd MacBeath-Simmons circles

  contributed by César Eliud Lozada, October 21, 2021. In a triangle ABC, let Ω₁ and Ω₂ be two circles, both passing through X₁₂₅ and with centers X₄₅₂₅₆ and X₄₅₂₅₇, respectively. Their respective squared radius are:

  ρ₁² = (\(\sqrt{3}\)*(S^2+(4*R^2-SW)^2)*S-(16*R^2-5*SW)*S^2+(4*R^2-SW)*(16*R^4-3*SW*(4*R^2-SW)))/(2*(SW+\(\sqrt{3}\)*S-R^2))^2

  ρ₂² = (-\(\sqrt{3}\)*(S^2+(4*R^2-SW)^2)*S-(16*R^2-5*SW)*S^2+(4*R^2-SW)*(16*R^4-3*SW*(4*R^2-SW)))/(2*(SW-\(\sqrt{3}\)*S-R^2))^2

  Some properties of these circles are:

  Ω₁ and Ω₂ pass through X₁₂₅ and X₁₃₇, both points on the nine-point circle of ABC. Therefore Ω₁, Ω₂ and the nine-point circle of ABC are coaxial. Ω₁ is the bicevian conic (or cyclocevian circle) of X₁₉₇₁₂ and X₄₁₈₉₈. Ω₂ is the bicevian conic (or cyclocevian circle) of X₁₉₇₁₃ and X₄₁₈₉₇. If ABC is acute then Ω₁ passes through the four intersections of MacBeath inconic (foci X₃ and X₄) and 1st Simmons inconic (foci X₁₃ and X₁₅). If ABC is obtuse then Ω₂ passes through the four intersections of MacBeath inconic and 2nd Simmons conic (foci X₁₄ and X₁₆).

  Ω₁ and Ω₂ are named here the 1st MacbBeath-Simmons circle and the 2nd MacbBeath-Simmons circle, respectively.

  Note: Simmons conics were introduced in the preamble just before X₄₁₉₉₃.

• X₄₅₂₆₆–X₄₅₂₈₅ — (B)-line conjugates

  contributed by Clark Kimberling and Peter Moses, October 21, 2021. If points P and U, given by trilinears P =\(p : q : r\) and U =\(u : v : w\) are distinct, and neither lies on a sideline BC, CA, AB, then the P-line conjugate of U is the point given by trilinears

  p(v^2 + w^2) - u(qv + rw) : q(w^2 + u^2) - v(rw + pu) : r(u^2 + v^2) - w(pu + qv).

  Now suppose that P and Q have barycentrics \(p : q : r\) and \(u : v : w\), and define the P-(B)line conjugate of U as the point F(P,U) given by barycentrics

  p(v^2 + w^2) - u(qv + rw) : q(w^2 + u^2) - v(rw + pu) : r(u^2 + v^2) - w(pu + qv).

  The points X₄₅₂₆₆ - X₄₅₃₈₅ are (B)line conjugates.

  The appearance of (i,j) in the following list means that X_i and X_j are on the Euler line and F(X_i) = X_j, so that also, F(X_j) = X_i:

  (2,30), {3,297), (4,441), (5,401), (20,44334), (30,2), (140,40853), (297,3), (376,44216), (381,40884), (384,21536), (401,5), (427,15013), [and others].

  See also the preambles just before X₄₄₈₈₆ and X₄₅₁₉₆.

• X₄₅₂₈₉–X₄₅₂₉₇ — 12th Vijay transforms

  contributed by Clark Kimberling and Peter Moses, October 24, 2021. The Vijay 12th parallel transform, V12, is defined in the preamble just before X₄₃₉₇₀ as follows: if U =\(u : v : w\) (barycentrics), then

  V12(U) = v*w^2 - 3*u*w^2 + v^2*w + 2*u*v*w + u^2*w - 3*u*v^2 + u^2*v : :

  Let KK denote the cubic that is the isotomic conjugate of K015. If X lies on infinity line or the Steiner circumellipse, then V12(X) lies on KK. The appearance of (i,j) in the following examples means that V12(X_i) = X_j:

  (30,45289), (512,44007), (513,44008), (514,44009), (519,20042), (522,45290), (523,44010), (524,45291), (525,45292), (527,45293), (690,45294), (900,48295).

• X₄₅₃₁₀–X₄₅₆₉₃ — and X₄₅₆₅₇-X₄₅₆₉₃, Midpoints of X₂ and other points

  contributed by Clark Kimberling and Peter Moses, October 31, 2021. Suppose that P =\(p : q : r\) and U =\(u : v : w\) (barycentrics). Then the P-(B):term:line conjugate <line-conjugate> of the line L at infinity is the point

  a*(v^2 + w^2) + (v + w)*(b*v + c*w) : :

  The locus of this point as U ranges through L is the ellipse, E(P,L), given by

  (q + r) x^2 + (r + p) y^2 + (p + q) z^2 - (q + r) y z - (r + p) z x - (p + q) x y = 0.

  The center of E(P,L) is the midpoint of X₂ and P, with barycentrics

  4 p + q + r : p + 4 q + r : p + q + 4r.

  The perspector E(P,L) is 1/(p^2 + 2 q^2 + 2 r^2 + 5 q r + p q + p r) : : .

  The major axis of E(P,L) is parallel to the major axis of the Steiner circumellipse, and the minor axis of E(P,L) is parallel to the minor axis of the Steiner circumellipse. Thus, the ellipse E(P,L) is simply and translation and dilation of the Steiner circumellipse.

• X₄₅₃₄₅–X₄₅₆₅₆ — Inverse triangles and anti-triangles

  contributed by César Eliud Lozada, October 30, 2021. Inverse triangles were introduced by Peter Moses in the preamble just before X₄₂₀₀₅ with more or less these terms: Suppose that T = A'B'C' is a triangle with vertices A', B', C' represented in normalized barycentric coordinates. Let M be the matrix representation of A'B'C', and let \(M^{-1}\) denote the inverse of M. Then the rows of \(M^{-1}\), interpreted as vertices of a triangle \(T^{-1}\), define the inverse triangle of A'B'C'.

  Matrices M and \(M^{-1}\) algebraically behave as expected; i.e., products of matrices M.\(M^{-1}\) = \(M^{-1}\).M = (3x3)-identity matrix, which represents ABC. Then it should be reasonable to expect that triangles T-of-\(T^{-1}\) and \(T^{-1}\)-of-T should be both ABC, in other words, that \(T^{-1}\) is the anti-triangle-of-T and vice-versa, but actually, this only occurs when T and ABC are similar or homothetic and, in both cases, T ‘ and T are homothetic triangles.

  The following properties can be deduced: [list and table omitted here].

• X₄₅₆₉₅–X₄₅₇₂₉ — Ortho-perspective and para-perspective triangles

  contributed by César Eliud Lozada, November 1, 2021.

  1. Let T₁=A₁B₁C₁ and T₂=A₂B₂C₂ be two non-orthologic triangles. Denote as T' = A'B'C' the triangle bounded by the lines through A₁, B₁, C₁ perpendicular to B₂C₂, C₂A₂, A₂B₂, respectively, and denote as T″ = A″B″C″ the triangle bounded by the lines through A₂, B₂, C₂ perpendicular to B₁C₁, C₁A₁, A₁B₁, respectively. Then for, some pairs of triangles T₁ and T₂, T₁ and T' are perspective if, and only if, T₂ and T″ are perspective.

  2. Let T₁=A₁B₁C₁ and T₂=A₂B₂C₂ be two non-parallelogic triangles. Denote as T' = A'B'C' the triangle bounded by the lines through A₁, B₁, C₁ parallel to B₂C₂, C₂A₂, A₂B₂, respectively, and denote as T″ = A″B″C″ the triangle bounded by the lines through A₂, B₂, C₂ parallel to B₁C₁, C₁A₁, A₁B₁, respectively. Then for, some pairs of triangles T₁ and T₂, T₁ and T' are perspective if, and only if, T₂ and T″ are perspective.

  In case (1), triangles T₁ and T₂ are said to be ortho-perspective and perspectors (T₁, T') and (T₂, T″) are named here the ortho-perspector T₁ to T₂ and ortho-perspector T₂ to T₁, respectively.

  In case (2), triangles T₁ and T₂ are said to be para-perspective and perspectors (T₁, T') and (T₂, T″) are introduced here as the para-perspector T₁ to T₂ and para-perspector T₂ to T₁, respectively.

  Reference: Vu Thanh Tung, Euclid 2834 (Message slightly modified by César Lozada).

• X₄₅₇₃₈–X₄₅₇₄₄ — Perspectors involving the Dao triangle

  This preamble, based on notes from Dao Thanh Oai, was contributed by Clark Kimberling and Peter Moses, Novermber 2, 2021, with amendments by Randy Hutson, November 7, 2021. Let (a) denote the A-Soddy ellipse, through A that has foci B and C. Define (b) and (c) cyclically. (The A-Soddy ellipse is defined by Randy Hutson at X₆₃₄₉. The A-Soddy hyperbola also passes through A and has foci B and C; see Paul Yiu, Introduction to the Geometry of the Triangle, p. 143). . Let

  d(a) = (b)∩(c); d(b) = (c)∩(a); d(c) = (a)∩(b); the lines d(a), d(b), d(c) concur in X₂₀.

  A' = d(a)∩BC; B' = d(b)∩CA; C' = d(c)∩AB; the lines AA″, BB″, CC″ concur in X₁₀₄₃.

  The triangle A'B'C', here named the Dao triangle, is perspective to certain other triangles, with perspectors X₄₅₇₃₈-X₄₅₇₄₄. (Actually, the Dao triangle is the cevian triangle of X₁₀₄₃; Randy Hutson, November 7, 2021)

  The conic (a),

  2*(a^2 + b^2 + 2*b*c + c^2)*y*z + 4*(b + c)*x*(c*y + b*z) - (a - b - c)*(a + b + c)*(y^2 + z^2) = 0,

  passes through the A-vertex of these triangles: anticomplementary, orthic-of-anticomplementary (also called the Gemini 113 triangle, the dual of the orthic triangle, the 1st anti-circumperp triangle, and the circumanticevian triangle of X₂).

  The line d(a) is given by

  (b - c)*(a + b + c)^2*x + (a + b - c)^2*(a + c)*y + (-a - b)*(a - b + c)^2*z = 0. The A-vertex of the Dao triangles is

  A' = 0 : (a + b)*(a - b + c)^2 : (a + b - c)^2*(a + c),

  The Dao triangle is perspective to every anticevian triangle; specifically, if P =\(p : q : r\), then the perspector is

  p ((a + b - c)^2*(a - b + c)^2*(b + c)*p - (a - b - c)^2*(a + b - c)^2*(a + c)*q - (a + b)*(a - b - c)^2*(a - b + c)^2*r) : :

  If P lies on the cubic K004, then A'B'C' is perspective to the antipedal triangle of P.

  The conics (b) and (c) meet in two real points, assumed in the discussion above, and also two nonreal points, considered here. In this case, the points d(a), d(b), d(c) are nonreal, and the vertices A', B', C', given by

  A' = 0 : -a - b : a + c, B' = b + a, 0, -b - c, C' = -c - a, c + b, 0,

  are collinear on this line:

  (b + c) x + (c + a) y + (a + b) z = 0,

  which passes through X_i for these i: 239, 514, 649, 1019, 1021, 3218, 3798, 4025, 4063, 4091, [and others].

• X₄₅₈₁₀–X₄₅₈₄₁ — CT-Perspectors

  contributed by César Eliud Lozada, November 7, 2021. Let T₁=A₁B₁C₁ and T₂=A₂B₂C₂ be two triangles.

  Tangents from A₁, B₁, C₁ to circles (A₁B₂C₂), (B₁C₂A₂), (C₁A₂B₂) bound a triangle T'=A'B'C'.

  Tangents from A₂, B₂, C₂ to circles (A₂B₁C₁), (B₂C₁A₁), (C₂A₁B₁) bound a triangle T″=A″B″C″.

  Then the following statements are equivalent:

  1. Triangles A₁B₁C₁ and A″B″C″ are perspective.

  2. Triangles A₂B₂C₂ and A'B'C' are perspective.

  3. (B₁A₂/C₁A₂)*.(C₁B₂/A₁B₂)*(A₁C₂/B₁C₂)=1.

  Reference: Vu Thanh Tung, Euclid 3001 (Message slightly modified by César Lozada).

• X₄₅₈₄₅–X₄₅₈₇₃ — Points associated with Vijay polar medial circles and Vijay polar medial triangles

  contributed by Clark Kimberling (Nov 3, 2021), based on notes from Dasari Naga Vijay Krishna, November 3, 2021. In the plane of a triangle ABC, let

  (O) = circumcircle of triangle ABC

  A'B'C' = medial triangle;

  (O)a = circle with B'C' as diameter, and define (O)b and (O)c cyclically;

  A_b = AB ∩ (O)a, and define B_c and C_a cyclically;

  A_c = AC ∩ (O)a, and define B_a and C_b cyclically;

  A₁ = A_bB' ∩ A_cC', and define B₁ and C₁ cyclically (here A₁, B₁, C₁ lies on the nine point circle of triangle ABC) ;

  K_a = polar of A wrt O_a (which passes through A₁), and define K_b and K_c cyclically;

  K'_a = polar of A' wrt (O)a, and define K'_b and K'_c cyclically;

  K’’a= polar of A₁ wrt (O)a (which passes through A), and define K’’b and K’’c cyclically;

  Kab = polar of A_b wrt (O)a respectively, and define Kbc and Kca cyclically;

  Kac = polar of A_c wrt (O)a respectively, and define Kba and Kcb cyclically;

  (O)’a = the circle with B_cC_b as diameter, and define (O)’b and (O)’c cyclically;

  L_a = polar of A wrt (O)’a, and define L_b and L_c cyclically;

  L'_a = polar of A' wrt (O)’a, and define L'_b and L'_c cyclically;

  L’’a= polar of A₁ wrt (O)’a, and define L’’b and L’’c cyclically;

  Lab = polar of A_b wrt (O)’c, and define Lbc and Lca cyclically;

  Lac = polar of A_c wrt (O)’b, and define Lba and Lcb cyclically;

  (O)’’a = circle with A_bA_c as diameter, and define (O)’’b and (O)’’c cyclically;

  T_a = polar of A wrt (O)’’a, and define T_b and T_c cyclically;

  T'_a = polar of A' wrt (O)’’a, and define T'_b and T'_c cyclically;

  T’’a= polar of A₁ wrt (O)’’a, and define T’’b and T’’c cyclically;

  Tab = polar of A_b wrt (O)’’a, and define Tbc and Tca cyclically;

  Tac = polar of A_c wrt (O)’’a, and define Tba and Tcb cyclically;

  M_a = perpendicular bisector of A_bA_c, and define M_b, M_c cyclically;

  M'_a = Perpendicular bisector of B_cC_b or B'C', and define M'_b, M'_c cyclically;

  N_a = polar of A11 wrt to (O)a = line passing through A11 parallel to BC, and define N_b and N_c cyclically;

  N'_a = polar of A12 wrt to (O)a = line passing through A12 parallel to BC, and define N'_b and N'_c cyclically;

  P_a = polar of A13 wrt to (O)’a = line passing through A13 parallel to BC, and define P_b and P_c cyclically;

  P'_a = polar of A14 wrt to (O)’a = line passing through A14 parallel to BC, and define P'_b and P'_c cyclically;

  Q_a = line passing through A15 parallel to BC, and define Q_b and Q_c cyclically;

  Q'_a = line passing through A16 parallel to BC, and define Q'_b and Q'_c cyclically;

  VPMC1 = circumcircle of 23rd Vijay polar medial triangle;

  VPMC2 = circumcircle of 24th Vijay polar medial triangle;

  VPMC3 = circumcircle of 25th Vijay polar medial triangle;

  VPMC4 = circumcircle of 26th Vijay polar medial triangle;

  Define 26 points by the following intersections: [list omitted here]

  Related circles are here named as follows: [list omietted here]

  Barycentric equations for above defined circles and lines: [list omitted here].

  Related triangles are here named as follows: [list omitted here].

  The triangle AnBnCn is here named the nth Vijay polar medial triangle, for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 .

  Barycentrics for points defined above: [list omitted here].

  Perspectors: [list omitted here].

• X₄₅₈₈₁–X₄₅₉₅₅ — Intersections of remarkable lines

  contributed by César Eliud Lozada, November 10, 2021.

  This section deals with the intersections of these remarkable lines: Antiorthic axis = X₄₄X₅₁₃ Fermat axis = X₆X₁₃ I-N line = X₁X₅ N-K line = X₅X₆ Apollonius line = X₁X₁₈₁ Gergonne line = X₂₄₁X₅₁₄ I-O line = X₁X₃ Orthic axis = X₂₃₀X₂₃₁ Brocard axis = X₃X₆ G-K line = X₂X₆ Lemoine axis = X₁₈₇X₂₃₇ Sherman line = X₃₂₅₉X₃₃₂₆ Brocard line = X₃₉X₅₁₂ Hatzipolakis axis = X₅X₅₂₃ Line at infinity = X₃₀X₅₁₁ Soddy line = X₁X₇ De Longchamps line = X₃₂₅X₅₂₃ I-H line = X₁X₄ Nagel line = X₁X₂ van Aubel line = X₄X₆ Euler line = X₂X₃ I-K line = X₁X₆ Napoleon axis = X₆X₁₇

  Intersections of these lines are showed in the following list: [long list omitted]. Center X₄₅₉₇₉, Points associated with Vijay orthic medial circles and Vijay orthic medial triangles,contributed by Clark Kimberling (Nov 15, 2021), based on notes from Dasari Naga Vijay Krishna, Nov 15, 2021.

  In the plane of a triangle ABC, let

  A'B'C' = medial triangle ;

  ApBpCp = orthic triangle ;

  (O)a = circle with B'C' as diameter, and define (O)b and (O)c cyclically;

  [Here (O)a, (O)b, (O)c are known as 1st Vijay A, B, C orthic medial circles; see the preamble just before X₄₅₈₄₅.]

  A_b = AB ∩ O_a, and define B_c and C_a cyclically;

  A_c = AC ∩ O_a, and define B_a and C_b cyclically;

  Define 8 points by the following intersections:

  A₁ = midpoint of A_bA_c, and define B₁ and C₁ cyclically;

  A₂ = midpoint of B_cC_a, and define B₂ and C₂ cyclically;

  A₃ = the point in (O)b ∩ (O)c other than A' and define B₃ and C₃ cyclically;

  A₄ = B_cB_a ∩ C_aC_b, and define B₄ and C₄ cyclically;

  A₅ = A_bB_c ∩ A_cC_b, and define B₅ and C₅ cyclically;

  A₆ = A_bC_a ∩ A_cB_a, and define B₆ and C₆ cyclically;

  A₇ = B_cC_a ∩ C_bB_a, and define B₇ and C₇ cyclically;

  A₈ = A_bC_b ∩ A_cB_c, and define B₈ and C₈ cyclically;

  A₉ = reflection of A₃ wrt to BC, and define B₉ and C₉ cyclically;

  A10 = midpoint of AAp and define B10 and C10 cyclically;

  Related conic here named as follows:

  There is a conic passes through the six points {A₃, B₃, C₃, A10, B10, C10} here named as Vijay orthic medial conic.

  This conic also contains the X₆₇₂₀[ and this is the only ETC center present on this conic] .

  The barycentric equation of this conic is given by [equation omitted here]. The Vijay orthic medial conic is an ellipse, a parabola, or a hyperbola according V is positive, zero, or negative, where [equation omitted here].

  The triangle AnBnCn is here named the nth Vijay orthic medial triangle, for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

  Barycentrics for points defined above: list omitted here].

  Perspectors: [list omitted here].

  X₄₅₉₇₉ = center of Vijay orthic medial conic

Part 24

• X₄₆₀₀₁–X₄₆₀₀₈ — Centers of Panchapakesan circles

  contributed by César Eliud Lozada, November 15, 2021. Let ABC be a triangle and P, Q two isogonal conjugate points (Q also denoted as \(P^{-1})\). Let P_a, P_b, P_c the centers of circles {{P,B,C}}, {{P,C,A}} and {{P,A,B}}, respectively, and similarly, let Q_a, Q_b, Q_c the centers of circles {{Q,B,C}}, {{Q,C,A}} and {{Q,A,B}}, respectively. Denote A_p = AP ∩ QQ_a and cyclically B_p and C_p. Finally, denote A_q=AQ ∩ PP_a and cyclically B_q and C_q. Then eight points P, Q, A_p, B_p, C_p, A_q, B_q, C_q lie on a circle. (Sriram Panchapakesan, Euclid 3162, November 14, 2021.)

  The above described circle is named here the Panchapakesan circle of P and Q. For P =\(x:y:z\), its center O* is:

  O* = a^2*(a^2*y^2 + b^2*x^2 + (a^2 + b^2 - c^2)*x*y)*(a^2*z^2 + c^2*x^2 + (a^2 - b^2 + c^2)*x*z)*(z^2*b^2 - y^2*c^2) : :

  Some properties of this circle are:

  O* lies in the infinity if P (or Q) lie in the infinity, or on the circumcircle or on McCay cubic K003. (César Lozada, Euclid 3174) If K(P, ABC) denotes the Panchapakesan circle of P and \(P^{-1}\), then the four circles K(P, ABC), K(A, BCP), K(B, CAP), K(C, ABP) are concurrent in a point R*(P). (Vu Thanh Tung, Euclid 3175) R*(P) is the circumcircle-inverse of \(P^{-1}\) . (César Lozada, Euclid 3181) The circumcircle-inverses of P and \(P^{-1}\) also lie on the Panchapakesan circle of P and \(P^{-1}\). (Elias M. Hagos , Euclid 3178) For any P, the circumcircle of ABC and the Panchapakesan circle of P and \(P^{-1}\) are orthogonal. (Sriram Panchapakesan, Euclid 3179)

  In the following list, (i, j) means that the center of the Panchapakesan circle of X_i and its isogonal conjugate is X_j: (2, 46001), (3, 523), (4, 523), (5, 46002), (6, 46001), (7, 46003), (8, 46004), (9, 46006), (11, 46007), (13, 6137), (14, 6138), (15, 6137), (16, 6138), (20, 46005), (36, 1769), (54, 46002), (55, 46003), (56, 46004), (57, 46006), (59, 46007), (64, 46005), (80, 1769), (186, 46008), (265, 46008), (1157, 46002), (1263, 46002).

• X₄₆₁₁₆–X₄₆₁₂₂ — Centers of Panchapakesan circles

  :contributed by César Eliud Lozada, November 15, 2021. Let ABC be a triangle and P, Q two isogonal conjugate points (Q also denoted as \(P^{-1})\). Let P_a, P_b, P_c the centers of circles {{P,B,C}}, {{P,C,A}} and {{P,A,B}}, respectively, and similarly, let Q_a, Q_b, Q_c the centers of circles {{Q,B,C}}, {{Q,C,A}} and {{Q,A,B}}, respectively. Denote A_p = AP ∩ QQ_a and cyclically B_p and C_p. Finally, denote A_q=AQ ∩ PP_a and cyclically B_q and C_q. Then eight points P, Q, A_p, B_p, C_p, A_q, B_q, C_q lie on a circle. (Sriram Panchapakesan, Euclid 3162, November 14, 2021.)

  The above described circle is named here the Panchapakesan circle of P and Q. For P =\(x:y:z\), its center O* is:

  O* = a^2*(a^2*y^2 + b^2*x^2 + (a^2 + b^2 - c^2)*x*y)*(a^2*z^2 + c^2*x^2 + (a^2 - b^2 + c^2)*x*z)*(z^2*b^2 - y^2*c^2) : :

  Some properties of this circle are:

  O* lies in the infinity if P (or Q) lie in the infinity, or on the circumcircle or on McCay cubic K003. (César Lozada, Euclid 3174) If K(P, ABC) denotes the Panchapakesan circle of P and \(P^{-1}\), then the four circles K(P, ABC), K(A, BCP), K(B, CAP), K(C, ABP) are concurrent in a point R*(P). (Vu Thanh Tung, Euclid 3175) R*(P) is the circumcircle-inverse of \(P^{-1}\) . (César Lozada, Euclid 3181) The circumcircle-inverses of P and \(P^{-1}\) also lie on the Panchapakesan circle of P and \(P^{-1}\). (Elias M. Hagos , Euclid 3178) For any P, the circumcircle of ABC and the Panchapakesan circle of P and \(P^{-1}\) are orthogonal. (Sriram Panchapakesan, Euclid 3179)

  In the following list, (i, j) means that the center of the Panchapakesan circle of X_i and its isogonal conjugate is X_j: (2, 46001), (3, 523), (4, 523), (5, 46002), (6, 46001), (7, 46003), (8, 46004), (9, 46006), (11, 46007), (13, 6137), (14, 6138), (15, 6137), (16, 6138), (20, 46005), (36, 1769), (54, 46002), (55, 46003), (56, 46004), (57, 46006), (59, 46007), (64, 46005), (80, 1769), (186, 46008), (265, 46008), (1157, 46002), (1263, 46002).

• X₄₆₁₃₇–X₄₆₁₉₀ — Centers related to bicentric pair PU(197)

  contributed by César Eliud Lozada, December 7, 2021. Bicentric points P(197) and U(197) are the real foci of the De Longchamps ellipse. (See definition of this ellipse in Wolfram’s Mathworld and coordinates of its real foci in PU(197) (BicentricPairs.html#P197)).

• X₄₆₁₉₁–X₄₆₂₆₃ — Centers related to bicentric pairs PU(160) to PU(175)

  Contributed by César Eliud Lozada, December 7, 2021.

• X₄₆₂₇₀–X₄₆₃₂₈ — Centers related to bicentric pairs PU(176) to PU(196)

  Contributed by César Eliud Lozada, December 8, 2021.

• X₄₆₃₈₀–X₄₆₃₉₃ — Crossdifferences: X₁X(k)

  contributed by Clark Kimberling and Peter Moses, December 21, 2021. The appearance of (j,k) in the following list means that the crossdifference of X₁ and X_j is X_k:

  (2,649), (3,650), (4,652), (5,654), (6,513), (7,657), (8,649), (9,513), (10,649), (11,654), (12,654), (19,656), (20,657), (21,661), (22,46380), (23,46381), (25,2522), (27,46382), (28,656), (29,822), (30,9404), (31,661), (32,1491), (33,652), (34,652), (35,650), (36,650), (37,513), (38,661), (39,659), (40,650), (41,2254), (42,649), (43,649), (44,513), (45,513), (46,650), (47,661), (48,656), (50,18116), (54,2600), (55,650), (56,650), (57,650), (58,661), (592,46383), (60,2610), (63,661), (65,650), (69,2484), (71,46384), (72,513), (73,652), (75,798), (76,46385), (77,657), (78,649), (79,9404), (80,654), (81,661), (82,8061), (83,46386), (84,14298), (85,46387), (86,798), (87,20979), (88,1635), (89,4893), (90,46388), (92,822), (99,46389), (100,1635), (102,46390), (103,46391), (104,46392), (108,46393), (109,46394)

  For every point U, the crossdifference of X₁ and U lies on the anti-orthic axis, X₄₄ X₅₁₃.

• X₄₆₅₈₅–X₄₆₅₉₆ — Euler Line Intercepts

  contributed by Clark Kimberling and Peter Moses, January 11, 2022. In the plane of a triangle ABC with circumcircle (O) and Euler line L, let U₁ be a point on (O), and let

  U₁ = an arbitrary point on (O), and T₁ = line tangent to (O) at U₁;

  U₂ = reflection of U₁ in L, and T₂ = line tangent to (O) at U₂;

  U₃ = antipode of U₁, and T₃ = line tangent to (O) at U₃;

  U₄ = antipode of U₂, and T₄ = line tangent to (O) at U₄;

  P₁ = T₁∩T₂; P₂ = T₂∩T₃; P₃ = T₃∩T₄; P₄ = T₄∩T₁.

  The points P₁, P₂, P₃, P₄ are the vertices of a parallelogram. The following table shows results for several choices of the point U₁. In the table, each index k represents the triangle center X_k. U₁ U₂ U₃ U₄ P₁ P₂ P₃ P₄ 74 477 110 476 46585 46616 15329 46608 98 842 99 691 7418 46609 11634 44823 100 1290 104 2687 13583 46610 14127 46611 101 2690 103 2688 46595 46596 105 2752 1292 2691 46586 46593 107 1304 1294 2693 46587 46614 46613 108 2766 1295 2694 46588 111 2770 1296 2770 46589 112 935 1297 2697 46592 46614 46594 46615 930 1291 1141 14979 46590 933 972 18401 945 46591 The midpoints of segments U₁U₂, U₂U₃, U₃U₄, U₄U₁ are vertices of another parallelogram, as represented by the following table [tabl omitted herej]..

  If you have Geogebra, you can download Three Parallelograms (Three%20Parallelograms.ggb). (The vertices are U₁, U₂, U₃, U₄; P₁, P₂, P₃, P₄; m1, m2, m3, m4. Drag D to vary the line OD; drag U₁ to vary the 12 vertices.)

• X₄₆₆₈₁–X₄₆₆₉₆ — Mutual-reflections conics

  contributed by César Eliud Lozada, January 14, 2022. Let T'=A'B'C' and T″=A″B″C″ be two triangles, A₁ the reflection of A' in B″C″ and A₂ the reflection of A″ in B'C'. Define B₁, B₂, C₁, C₂ cyclically. For some selected pairs of triangles T' and T″, these six points lie on a conic, here named the mutual-reflections conic of T’ and T” or, shortly, the MR-conic of T’ and T”.

  The appearance of (T', T″, n) in the following partial list means that the center of the MR-conic of triangles T' and T″ is X_n: (ABC, ABC-X3 reflections, 3), (ABC, anticomplementary, 3), [and others].

• X₄₆₇₀₆–X₄₆₇₂₆ — Points on cubics

  :contributed by Clark Kimberling and Peter Moses, January 18, 2022. Let P =\(p : q : r\) be a point that is not on a sideline BC, CA, AB of a triangle ABC. Let A' be the point, other than A, in which the cevian line AP intersects the Steiner circumellipse, and define B' and C' cyclically. Then

  A' = -a q r : (b r + c q) q : (b r + c q) r

  Let A″ be the points, other than A, in which the anticevian line of P intersects the Steiner circumellipse, and define B″ and C″ cyclically. Then

  A″ = a q r : (c q - b r) q : (b r - c q) r

  Then A'B'C' is perspective to the anticomplementary triangle, and the perspector is the point

  U(P) = p^2*q^2 + p^2*r^2 - q^2*r^2 : :

  Moreover, A″B″C″ is also perspective to the anticomplementary triangle, with perspector U(P), which is the anticomplement of isotomic conjugate of P^2. The appearance of (i,j) in the following list means that if X_i =\(p:q:r\), then X_j = U(P).

  (1,194), (2,2), (4,6392), (6,8264), (7,4452), (8,30695), [and others].

  Note that i = j if and only if X_i lies on the Steiner circumellipse.

• X₄₆₇₃₈–X₄₆₇₅₉ — Points on cubics pK(m^2 n^2, m^2 n^2 q r)

  contributed by Clark Kimberling and Peter Moses, January 24, 2022. In the plane of a triangle ABC, let L, given by l x + m y + n z = 0, be a line, and let P =\(p : q : r\) be a point not on L and not on a sideline BC, CA, AB. Let

  Ap = L∩AP, Bp = L∩BP, Cp = L∩CP.

  Let A'B'C' be the cevian triangle of a point X =\(x : y : z\), so that

  A' =\(0 : y : z\) , B' =\(x : 0 : z\), C' =\(x : y : 0\).

  The locus of X such that A'B'C' is perspective to the (degenerate) triangle ApBpCp is the cubic pK(m^2 n^2, m^2 n^2 q r), given by

  p (m^2 q y^2 z - n^2 r y z^2) + (cyclic) = 0.

  The appearance of {{i,j}, {{Knnn}, {\(h_{1}\), \(h_{2}\), …}}} in the following list means that if \(l : m : n\) = X_i and P = X_j, then the cubic is Knnn in Bernard Gibert’s catalogue (CTC), and that the cubic passes through the points \(X(h_{1})\), \(X(h_{2})\), …

  {{1,1},{{},{1,75,92,304,561,1760,46244}}}

  {{1,2},{{K184},{2,69,75,76,85,264,312,15466,34403,34404,40702}}}

  {{1,4},{{},{2,4,75,76,253,305,341,1088,1370,14615,20914,40009,40015,46738,46739,46740,46741}}}

  {{1,10},{{},{2,10,75,76,310,4043,17135,40004,40005}}}

  [and other]

  Each of these cubics is unicursal with node and singularity m n : n l : l m. As noted above, each cubic is of type pK(m^2 n^2, m^2 n^2 q r), which is equivalent to cK(# m n , m^2, n^2 q r). See Special Isocubics in the Triangle Plane). (Bernard Gibert, January 28, 2022)

• X₄₆₇₇₉–X₄₆₈₁₅ — Points on cubics

  :contributed by Clark Kimberling and Peter Moses, January 27, 2022. In the plane of a triangle ABC, let L, given by l x + m y + n z = 0, be a line, and let P =\(p : q : r\) be a point not on L and not on a sideline BC, CA, AB. Let

  Ap = L∩AP, Bp = L∩BP, Cp = L∩CP.

  Let A'B'C' be the cocevian triangle (TCCT, p. 200) of a point X =\(x : y : z\), so that

  A' =\(0 : y : -z\), B' = \(-x : 0 : z\), C' =\(x :-y : 0\).

  The locus of X such that A'B'C' is perspective to the (degenerate) triangle ApBpCp is the cubic, given by

  m^2 p q y^2 z + n^2 p r y z^2 + (cyclic) - 2(m n q r + n l r p + l m p q) x y z = 0.

  The appearance of (k, Knnn) in the following list means that if L is the line \(x + y + z = 0\) at infinity and P = X_k, then the cubic is Knnn in Bernard Gibert’s catalogue (CTC):

  (2, K015), (4, K010), (99, K185), (190, K296), (648, K953), (876, K286), (4373, K090), (5485, K408)

  The appearance of {{i,j}, {h1, h2, …}}} in the following list means that if \(l : m : n\) = X_i and P = X_j, then the cubic passes through the points \(X(h_{1})\), \(X(h_{2})\), …

  {1,{2,1022,1026,3572,4585,24004,24015,24029,24035,27853,46779,46780,46781,46782,}

  {76,{2,4230,17941,23288,23342,23354,30508,30509}

  {100,{2,88,3218,3935,4358,18359,37780,37783,41798}

  [and others].

• X₄₆₈₂₆–X₄₆₈₄₄ — Points on cubics

  contributed by Clark Kimberling and Peter Moses, undated. In the plane of a triangle ABC, let L, given by l x + m y + n z = 0, be a line, and let P =\(p : q : r\) be a point not on L and not on a sideline BC, CA, AB. Let

  Ap = L∩AP, Bp = L∩BP, Cp = L∩CP.

  Let A'B'C' be the generalized Gemini triangle 34 a point X =\(x : y : z\); that is,

  A' = \(-x : z : y\), B' =\(z : -y : x\), C' =\(y : x : -z\) .

  The locus of X such that A'B'C' is perspective to the (degenerate) triangle ApBpCp is the cubic, given by

  l p (n r - m q) x^3 - (l m p r + m^2 q r + l^2 p q + m n p q) y^2 z + (l n p q + n^2 q r + l^2 p r + m n p r) y z^2 + (cyclic) = 0.

  This section exemplifies the cubic for three cases:

  Case 1: L =\(1 : 1 : 1\) (the line at infinity). Here, the cubic is given by

  p (r - q) x^2 - (p r + q r + 2 p q) y^2 z + ) p q + q r + 2 p r) y z^2 + (cyclic) = 0.

  Case 2: P =\(1 : 1 : 1\) (centroid). Here, the cubic is given by

  l (n - m) x^3 - (m^2 + l^2 + l m + m n) y^2 z + (n^2 + l^2 + l n + m n) y z^2 + (cyclic) = 0. Case 3: P =\(1 : 1 : 1\) (centroid). Here, the cubic is given by

  a^2 ( c^2 - b^2) x^3 - (a^2 b (a + c) + b^2 c (a + b)) y^2 z + (a^2 c (a + b) + b c^2 (a + c)) y z^2 = 0.

  Examples for Cases 1, 3, and 3 are omitted here.

• X₄₆₈₇₇–X₄₆₈₈₈ — Points on cubics

  :Let Ap, Bp,Cp be as in the preambles just before X₄₆₇₃₈ and X₄₆₇₇₉. The locus of a point X =\(x : y : z\) such that the anticevian triangle of X is perspective to the triangle ApBpCp is given by the following cubic:

  p(m q + n r)(m y^2 z - n y z^2) + (cyclic) = 0,

  which, in the Gibert classification system, is pK(-m n p (m q + n r) : : , m n : : ).

  In the following list, the appearance of {j,{Knnn},{{h1, h2, …}} means that if P = X₁ and L is given by l x + m y + n z = 0, Where \(l : m : n\) = X_j, then the cubic passes through the points \(X(h_{1})\), \(X(h_{2})\), …

  {2,{K345},{1,2,9,10,37,226,281,1214,7952,39131}}

  {4,{},{1,69,72,306,1439,1763,7289,17170,17441}}

  [and others].

  In the following list, the appearance of {j,{Knnn},{{h1, h2, …}} means that if P = X₂ and L is given by l x + m y + n z = 0, Where \(l : m : n\) = X_j, then the cubic passes through the points \(X(h_{1})\), \(X(h_{2})\), …

  {1,{K366},{2,8,10,75,307,318,321,1441}}

  {3,{K674},{2,4,5,264,311,324,847,39113,39114,39115,39116,39117}}

  {4,{K099},{2,3,20,63,69,77,78,271,394,7013,15394,46351}}

  [and others]

  In the following list, the appearance of {j,{Knnn},{{h1, h2, …}} means that if P = X₆ and L is given by l x + m y + n z = 0, Where \(l : m : n\) = X_j, then the cubic passes through the points \(X(h_{1})\), \(X(h_{2})\), …

  {2,{K836},{2,3,6,39,141,427,5403,5404,14376,40938}}

  {69,{K350},{4,5,6,25,51,52,53,14593}}

  {75,{K362},{1,6,33,37,42,55,65,73,2331,41086,41087,41088}}

  [and othersj].

  In the following list, the appearance of {j,{Knnn},{{h1, h2, …}} means that if P = X₇₅ and L is given by l x + m y + n z = 0, Where \(l : m : n\) = X_j, then the cubic passes through the points \(X(h_{1})\), \(X(h_{2})\), …

  {6,{},{75,76,312,313,321,349,1231,7017}}

  {9,{},{2,7,75,85,4146,18743,27818,27828}}

  {19,{},{63,69,75,304,326,345,348,18750,44189}}

  [and others].

  In the following list, the appearance of {j,{Knnn},{{h1, h2, …}} means that if P = X₇₆ and L is given by l x + m y + n z = 0, Where \(l : m : n\) = X_j, then the cubic passes through the points \(X(h_{1})\), \(X(h_{2})\), …

  {3,{},{2,4,76,264,491,492,34208}}

  {20,{},{4,76,253,459,9307,13567,41005}}

  {21,{},{10,76,226,1211,1441,18697,45196}}

  [and others].

• X₄₆₈₉₄–X₄₆₉₁₉ — Type 1 Pappus points

  contributed by Clark Kimberling and Peter Moses, February 9-10, 2022. In the plane of a triangle ABC, suppose that L₁, given by l1 x + m1 y + n1 z = 0, is a line, and P₁ =\(p1:q1:r1\) is a point not on L₁ and not on a sideline BC, CA, AB. The cevians of P₁ are the lines AP₁, BP₁, CP₁, for which coefficients are

  \(0 : q : r\),\(p : 0 : r\),\(p : q : 0\), respectively. Let

  Ap = L₁∩AP₁, Bp = L₂∩BP₁, Cp = L₂∩CP₁

  Similarly, suppose that L₂, given by l2 x + m2 y + n2 z = 0, is another line, and that P₂ =\(p2:q2:r2\) is a point not on L₂ and not on a sideline BC, CA, AB. Let

  Au = L₂∩AP₂, Bu = L₂∩BP₂, Cu = L₂∩CP₂

  By Pappus’s Theorem, the points

  BpCu∩CpBu, CpAu∩ApCu, ApBu∩BpAu

  are collinear (on the Pappus line of the configuration). The Type 1 Pappus Point of L₁, P₁, L₂, P₂ is the point whose barycentrics are coefficients of the Pappus line. Barycentrics for this point follow:

  l2 p2 q1 r1 (m1 q2 + n1 r2) + l1 p1 q1 r2 (l2 p2 + n2 r2) + l1 p1 q2 r1 (l2 p2 + m2 q2) : :

• X₄₆₉₆₁–X₄₆₉₇₀ — Perspectrors of circumcevian triangles and their inverses

  contributed by Clark Kimberling and Peter Moses, February 20, 2022.For an introduction to inverse triangles, see the preambles just before X₄₂₀₀₅, X₄₃₂₈₀, and X₄₃₃₄₃. This present section extends the section consisting of X₄₃₃₄₄-X₄₃₃₆₃.

  Let O = X₃ = circumcenter fo ABC, and suppose that P is a point not on a sideline BC, CA, AB. Let T be the circumcevian triangle of P, and let T' be the inverse of T, as defined in the preamble just before X₄₂₀₀₅. Then T and T' are perspective, and the perspector is the point on the circumcircle whose isogonal conjugate is the infinite point of intersection of all the lines perpendicular to OP. (Francisco Javier García Capitán , February 19, 2022)

  Continuing, let U be the isogonal conjugate just mentioned. The line OP meets the infinity line in the orthogonal conjugate of U. For a discussion of orthogonal conjugates, see Clark Kimberling, “Polynomial triangle centers on the line at infinity”: Journal of Geometry 111, Article 10)

  The appearance of (h,j,k) in the following list means that X_j is the perspector of the circumcevian triangle of X(h) and its inverse triangle, as defined near in the preamble just before X₄₂₀₀₅, and that X_j is the trilinear pole of X₆X_j.

  (1,100,1), (2,110,3), (4,110,3), (5,110,3), (6,99,2), (7,43344,13404), (8,901,101), (9,934,57), (10,109,41), (11,6099,906), (12,43345,8071), (13,10409,2981), (14,10410,6151), (15,99,2), (16,99,2), (17,10409,2981), (18,10410,6151), (19,13395,1214), (20,110,3), (21,110,3), (22,110,3), (23,110,3), (24,110,3), (25,110,3), (26,110,3), (27,110,3), (28,110,3), (29,110,3), (30,110,3), [and others]

• X₄₇₀₀₆–X₄₇₀₂₅ — Infinity-Incircle-transform

  contributed by César Eliud Lozada, February 26, 2022. Let ABC be a triangle, A_iB_iC_i the intouch triangle of ABC and P a point in the line at infinity. Let A'B'C' be the reflection of A_iB_iC_i in the line X₁P. Then A'B'C' is perspective to ABC and Q(P), the isogonal-conjugate of the perspector, lies on the incircle of ABC.

  The point Q is named here the infinity-incircle-transform of P. If P =\(x : y : z\) (barycentrics) then

  Q(P) = a^2*(a - b + c)*(a + b - c)*(c*y - b*z)^2 : :

  The appearance of (i, j) in the following list means that Q( X_i ) = X_j:: (30, 3024), (511, 3023), (512, 3027), (513, 1317), (514, 1362), (515, 1364), (516, 3022), (517, 11), (518, 1358), (519, 1357), (520, 3324), (521, 1359), (522, 1361), (523, 3028), (524, 3325), (525, 3320), [and others].

• X₄₇₀₉₀–X₄₇₁₀₁ — Shinagawa-Euler Points

  based on notes by Kiminari Shinagawa, contributed by Clark Kimberling and Peter Moses, March 8, 2022. The Euler coordinate system consists of an x-axis (the Euler line) and y-axis (the orthic axis). The axes meet in the origin, X₄₆₈ = (0,0). Points on the x-axis and on the same side of (0,0) as the centroid, G, have positive x-coordinates. Points on the y-axis and on the same side of (0,0) as the bicentric point P(201) have positive y-coordinates.

  A point with barycentrics \(p : q : r\) has Euler coordinates (x,y) given by

  x = (SA*p + SB*q + SC*r)/(p+q+r)

  y = (SA(SB-SC)*p + SB(SC-SA)*q +SC(SA - SB)*r)/(p+q+r).

  To convert from Euler coordinates (x,y) to barycentrics \(p : q : r\), let

  S^2 = SB*SC + SC*SA + SA*SB = 4*(area of ABC)^2

  E = (SB + SC)(SC + SA)(SA + SB)/S^2

  F = SA*SB*SC/S^2.

  Then

  p = (E+F)*SB*SC - 3F S^2 - (3SB*SC-S^2)x + (SB-SC)y

  q = (E+F)*SC*SA - 3F S^2 - (3SC*SA-S^2)x + (SC-SA)y

  r = (E+F)*SA*SB - 3F S^2 - (3SA*SB-S^2)x + (SA-SB)y

  Note that x must have degree 2 in a,b,c, and y must have degree 4.

  The appearance of (x,0;n) in the following list means that the point with Euler coordinates (x,0) is X_n:

  {(E/2,0); 10257}, {(-E/2,0); 37971}, {(E+F,0); 858}, {(2(E+F),0); 46517}, {(3(E+F),0); 5189}, {((E+F)/4,0);37911}, {(E-F,0); 2071}, {(F,0); 403}, {(2F,0); 10151}, {(4F,0); 13473}, {(F/2,0); 37942}, {(F/3,0); 37779}, {(-F,0); 186}, {(-2F,0); 37931}, {(-F/2,0); 37935}

  The appearance of (\(\$f(a,b,c)\$\),0;n) in the next list means that the point with Euler coordinates (f(a,b,c)+f(b,c,a)+f(c,a,b),0) is X_n, and that the point lies on the positive x-axis for all non-equilateral triangles:

  {(\(\$a^{2}\$\),0); 46517}, {(\(\$a^{2}\$\)/4,0); 5159}, {(\(\$a^{2}\$\)/6,0); 2}, {(\(\$aSA\$\)/\(\$a\$\),0);3109}}

  For more examples, see Euler Coordinates (Euler%20Coordinates.pdf).

  See also the preambles just before X₄₇₃₃₂ and X₄₇₄₈₈.

• X₄₇₁₂₂–X₄₇₁₉₈ — Points on the orthic axis

  contributed by Clark Kimberling and Peter Moses, March 12, 2022. This section consists of five types of points on the orthic axis:

  1. Points X₄₇₁₂₂-X₄₇₁₃₉. Suppose that X =\(x : y : z\), and let D(X) = SB z - SC y : : . Then D(X) lies on the orthic axis; indeed,

  D(X) = crossdifference of every pair of points on the line X₃X*, where X* = isogonal conjugate of isotomic conjugate of X. (Note that SB z - SC y : : lies on the orthic axis, whereas SB y - SC z : : lies on the line at infinity.

  The appearance of {h,k} in the following list means that X_k = D(X(h)): {1,6590}, {2,523}, {3,2501}, {4,647}, {5, 47122}, {6,523}, {7,650}, {8,650}, {9,47123}, {10,45745}, [and others].

  2. Points X₄₇₁₄₀-X₄₇₁₉₂. The appearance of {h,k} in the following list means that the line through X(h) parallel to the Euler line meets the orthic axis in X_k:

  {1,16272}, {6,16303}, {8,16304}, {10,16305}, {11,47140}, {12,47160}, {13,47141}, {14,47142}, {19,47161}, {32,16306}, {37,16307}, {39,16308}, {40,16309}, [and others].

  3. Points X₄₇₁₉₃-X₄₇₁₉₅. The appearance of {i, j; k} in the following list means that X_k = X_i-line conjugate of X_j:

  {230, 231; 47193}

  {230, 232; 47194}

  {230, 2501; 47195}

  {231, 230; 46953}

  {232, 230; 46953}

  {230, 647; 468}

  {647, 230; 46953}

  [and others].

  4. Points X₄₇₁₉₆-X₄₇₁₉₈. The appearance of {h, i; j, k} in the following list means that X_k = {X_i, X_j-harmonic conjugate of X(h), so that also, X(h) = {X_i, X_j-harmonic conjugate of X_k: {230,231,232 47196}, {230,231,468,47197}, {230,468, 647,47198}

  5. Points X₄₇₁₉₉-X₄₇₂₃₆. Four circles with center on the orthic axis are these: Stevanovic circle, Dao-Moses-Telv circle, Moses radical circle, and Moses-Parry circle. If P is a point on the orthic axis and (O) is a circle with center on the orthic axis, then the (O)(-inverse of P is also on the orthic axis, as exemplified by X₄₇₁₉₉-X₄₇₂₃₆.

• X₄₇₃₃₂–X₄₇₃₄₂ — Shinagawa-Euler points

  contributed by Peter Moses, Kiminari Shinagawa, and Clark Kimberling, April 1, 2022. Suppose that P and U are points, neither on the line at infinity, and that normed (i.e., “normalized”) barycentrcis are given by P= (p,q,r) and U = (u,v,w) Then the point P-U defined by the combo p-u : q - v : r - w is on the line at infinity. Let T(P,U) denote the point with Euler coordinates (x,y) given by

  x = SA*(p-u) + SB*(q - r) + SC*(r - w)

  y = SA*(SB - SC)*(p-u) + SB*(SC - SA)*(q - r) + SC*(SA - SB)*(r - w)

  If P and U are on the Euler line, then y = 0, and P - U = X₃₀, and T(P,U) lies on the Euler line. The appearance of (i,j;k) in the following list means that T(X_i,X_j) = X_k.

  (2,3; 47332), (2,4; 47031), (2,5; 18579), (2,23; 47311), (2,376; 47310), (2,381; 47333), (3,2; 47333), (3,4; 47308), (3,5; 47335), (3,20; 47309), (3,140; 18571), (3,186; 10257), (3,376; 47332), (3,381; 47031), (4,2; 47310), (4,3; 47309), (4,5; 47336), (4,23; 47339), [and others].

  See also the preambles just before X₄₇₀₉₀, X₄₇₄₈₈, and X₄₇₆₂₉.

• X₄₇₃₇₁–X₄₇₃₉₈ — Two-parallels points

  contributed by César Eliud Lozada, March 27, 2022. Let AA₁, BB₁, CC₁ be the altitudes in acute triangle ABC, and let X be an arbitrary point. Let M, N, P, Q, R, S be the feet of the perpendiculars from X to the lines AA₁,BC,BB₁,CA,CC₁,AB. Prove that MN, PQ, RS are concurrent. (Source: AOPS 713159, Dec 29, 2006.)

  The above can be generalized as follows: Let ABC be a triangle, X a point and P another point not on the side lines of ABC or on the cevian lines of X. The parallel lines through P to AX and BC cut BC and AX at A' and A″, respectively; B', B″ and C', C″ are built cyclically. Then the lines A'A″, B'B″, C'C″ concur.

  The point of intersection Q(X, P) of A'A″, B'B″, C'C″ is named here the X-two parallels point of-P. If X =\(X : Y : Z\) and P =\(x : y : z\) (both barycentrics) then:

  Q(X, P) = x*(-Y*Z*x^2 + (Y + Z)*X*y*z + (Y*z + Z*y)*X*x) : :

  Some properties: Q(X, P) is the midpoint of P and the X-Ceva conjugate of P. (Randy Hutson, March 28, 2022) For P on the line at infinity, Q(X, P) = P. For X=X₂, Q(X,P) is W(P), the center of the parallel-conic described above X₁₀₀₀₁. If X=X₄ and P lies on the nine-point-circle of ABC, then Q(X,P) is the complementary conjugate of P.

  The appearance of (i, j, k) in the following lists means that X_i- two parallels point of -X_j is X_k: [long lists omitted here].

• X₄₇₄₀₅–X₄₇₄₃₄ — Points on the bicevian conic of X₃ and X₆

  contributed by Clark Kimberling and Peter Moses, undated. Let BCC denote the bicevian conic of X₃ and X₆, which is given by the equation

  b^4*c^4*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*x^2 - 2*a^2*b^2*c^6*(a^2 + b^2 - c^2)*x*y - a^4*c^4*(a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*y^2 - 2*a^2*b^6*c^2*(a^2 - b^2 + c^2)*x*z + 2*a^6*b^2*c^2*(a^2 - b^2 - c^2)*y*z - a^4*b^4*(a^2 - b^2 - c^2)*(a^2 - b^2 + c^2)*z^2 = 0.

  This conic passes through the points X_i for these i: 3269, 7117, 9475, 15166, 15167, 20728, 20830, 20975, 22084, 22428. Bernard Gibert has noted that if P is a point on the line at infinity, then the X₃-Ceva conjugate of the barycentric product X₆*P is on BCC. (Equivalently, the X₆-Ceva conjugate of X₃*P is on BCC.) The line of P and the X₃-Ceva conjugate of X₆*P is tangent to the Steiner inellipse, which is bitangent to BCC. Moreover, if X is on the nine-point circle, then the isogonal conjugate of the polar conjugate of X is on BCC.

• X₄₇₄₈₈–X₄₇₅₀₈ — Points T(P,U) for P and U on selected lines

  contributed by Peter Moses, Kiminari Shinagawa, and Clark Kimberling, April 7, 2022. As in the preamble just before X₄₇₄₆₈, suppose that P and U are points, neither on the line at infinity, and that normed (i.e., “normalized”) barycentrics are given by P= (p,q,r) and U = (u,v,w) Then the point P-U defined by the combo p-u : q - v : r - w is on the line at infinity. Let T(P,U) denote the point with Euler coordinates (x,y) given by

  x = SA*(p-u) + SB*(q - r) + SC*(r - w)

  y = SA*(SB - SC)*(p-u) + SB*(SC - SA)*(q - r) + SC*(SA - SB)*(r - w)

  The appearance of (i,j;k) in the following lists means that T(X_i,X_j) = X_k:

  P and U on the Nagel line, X₁X₂:

  (1,2; 47472), (1,8; 47489), (1,10; 47491), (1,145; 47490), (1,551; 47495)

  (2,1; 47488), (2,8; 47493); (2,10;47495), [and others].
  P and U on the IK line, X₁X₆:

  (1,6; 47477), (1,9; 47507), (6,1; 47506), (9,1; 47508)

  P and U on the Euler line, X₂X₃: see the preamble just before X₄₇₃₃₂.

  P and U on the GK line, X₂X₆:

  (2,6;47473), (2,69; 47541), (2,81; 47542), (2,86; 47543), (2,141;47544)

  (6,2;47545), (6,69;47546), (6,81; 47547), [and others].
  P and U on the Brocard axis, X₃X₆:

  (3,6; 47468), (3,32; 47567), (3,39; 47568), (3,182; 47569), (3,187; 47570)

  (6,3; 47571), (6,32; 47572), (6,39; 47573), (6,187; 47574), [and others].
  P and U on the anti-orthic axis, X₄₄X₅₁₃:

  (649,650; 47499), (650,649; 47500)
  P and U on the Lemoine axis, X₁₈₇X₂₃₇:

  (182,237;47502), (237,187; 47501), (237,647; 47504), (647,187; 47503); (647,237; 47505)

  Note that the transformation T is many-to-one; i.e., for each T(P,U), there are (infinitely) many pairs (P',U') such that T(P',U') = T(P,U). For example, T(X₂,X₅₅₁) = T(X₁₀,X₂) = TX(551,X₁) = X₄₇₅₆₂, as listed above.

  See also the preambles just before X₄₇₀₉₀ and X₄₇₃₃₂.

• X₄₇₆₂₉–X₄₇₆₃₂ — Shinagawa-Euler points (k*Sω, 0) AND (k*S, 0)

  contributed by Peter Moses, Kiminari Shinagawa, and Clark Kimberling, April 21, 2022. See the preambles just before X₄₇₀₉₀, X₄₇₃₃₂, and X₄₇₄₈₈.

  The appearance of (k,n) in the following list means that the point having Euler coordinates (k*\(S_{\omega }\), 0) is X_n:

  (-3,37900), (-2,37899), (-1,23), (-3/4,47630), (-2/3,37904), (-1/2,37897), (-1/3,7426), (-1/4,47316), (1/4,37911), (1/3,2), (1/2,5159), (2/3,47097), (3/4,47629), (1,858), (2,46517), (3,5189)

• X₄₇₆₅₀–X₄₇₇₂₉ — Points in a [L(31),L(32)]-coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then the [L₁,L₂]-coordinate system is here defined as a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = L(31) = X₅₁₄X₆₆₁ = trilinear polar of (75);

  L₁ is given by the equation a α + b β + c γ = 0.
  L₂ = L(32) = X₃₂₅X₅₂₃ = deLongchamps axis = trilinear polar of X₇₆;

  L₂ is given by the equation a^2 α + b^2 β + c^2 γ = 0.

  The origin is given by (0,0) = X₆₉₃ = b c (b + c) : c a (c + a) : a b (a + b) = isotomic conjugate of X₁₀₀

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b-c)(bc + x - (b+c) y) : (c-a)(ca + x - (c+a)y) : (b-c)(ab + x - (a+b)y),

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 2, and y is symmetric and homogeneous of degree 1. Note that these results depend on the repesentations of L₁ and L₂ as shown, and that the systems [L₂,L₁] and [L₁,L₂] differ.

  The appearance of (x,y), k in the following table means that (x,y) = X_k: [table omitted here].

• X₄₇₇₅₄–X₄₇₈₄₅ — Points in a [X₂X_513, L(2)L(514)]-coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then the {L₁,L₂}-coordiinate system is here defined as a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = X₂X₅₁₃, given by the equation (2a-b-c)α + (2b-c-a)β + (2c-a-b)γ = 0.

  L₂ = X₂X₅₁₃, given by the equation (2bc-ca-ab)α + (2ca-ab-bc)β + (2ab-bc-ca)γ = 0.

  The origin is given by (0,0) = X₂ =\(1 : 1 : 1\)

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b-c)((a-b)(a-c) + x + ay) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 2, and y is symmetric and homogeneous of degree 1.

  The appearance of (x,y), k in the following table means that (x,y) = X_k: [table omitted here].

• X₄₇₉₀₀–X₄₈₁₅₁ — Points in the {X₂X_514, X₂X_523}-coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then an {L₁,L₂}-coordiinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁: (2a-b-c)α + (2b-c-a)β + (2c-a-b)γ = 0.

  L₂: (2a^2 - b^2 - c^2)α + (2b^2 - c^2-a^2)β + (2c^2 - a^2 - b^2)γ = 0.

  The origin is given by (0,0) = X₂ =\(1 : 1 : 1\)

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b-c)((a-b)(a-c) + x + (b+c)y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogenous of degree 2, and y is symmetric and homogeneous of degree 1.

  The appearance of {x,y}, k in the following table means that (x,y) = X_k: [table omitted here].

Part 25

• X₄₈₁₅₆–X₄₈₂₅₄ — Points in a [X₂X_513, X₂X_523] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁: (2bc - ca - ab)α + (2ca - ab - bc)β + (2ab - bc - ca)γ = 0.

  L₂: (2a^2 - b^2 - c^2)α + (2b^2 - c^2 - a^2)β + (2c^2 - a^2 - b^2)γ = 0.

  The origin is given by (0,0) = X₂ =\(1 : 1 : 1\).

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) ((a - b)(a - c)(a + b + c) + a x + (b + c) y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 2, and y is symmetric and homogeneous of degree 1.

  The appearance of {x,y}, k in the following table means that (x,y) = X_k:

  {-2 (a b+a c+b c), -2 (a b+a c+b c)},47775

  {-((2 a b c)/(a+b+c)), -((2 a b c)/(a+b+c))}, 47793

  {-2 (a b+a c+b c), 0}, 47821

  {-2 (a^2+b^2+c^2), a^2+b^2+c^2}, 31131

  {-2 (a b+a c+b c), a^2+b^2+c^2}, 30565

  [and others].

• X₄₈₂₆₄–X₄₈₂₈₀ — Points in a [X₅₁₄X_661, X₅₂₃X_661] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁: a α + b β + c γ = 0.

  L₂: (a + b)(a + c) α + (b + c)(b + a) β + (c + a)(c + b)γ = 0.

  The origin is given by (0,0) = X₆₆₁ = b^2 - c^2 : c^2 - a^2 : a^2 - b^2.

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) (a b + a c - x + (b + c)y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 2, and y is symmetric and homogeneous of degree 1.

  The appearance of {x,y}, k in the following table means that (x,y) = X_k:

  {-2 (a^2+b^2+c^2), -((2 (a^2+b^2+c^2))/(a+b+c))}, 48020

  {-2 (a b+a c+b c), -((2 (a b+a c+b c))/(a+b+c))}, 48021

  {-2 (a+b+c)^2, -2 (a+b+c)}, 48019

  {-2 (a^2+b^2+c^2), -((a^2+b^2+c^2)/(a+b+c))}, 47943

  [and others].

• X₄₇₂₈₁–X₄₈₃₀₇ — Points in a [X₁X_514, X₁X_523] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁: (b^2 + c^2 - a b - a c) α + (c^2 + a^2 - b c - b a) β + (a^2 + b^2 - c a - cb) γ = 0.

  L₂: (b^3 + c^3 - a^2 b - a^2 c) α (c^3 + a^3 - b^2 c - b^2 a) β (a^3 + b^3 - c^2 a - c^2 b) γ = 0.

  The origin is given by (0,0) = X₁ =\(a : b : c\).

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) (a (a - b)(a - c) - x + (b + c)y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 3, and y is symmetric and homogeneous of degree 2.

  The appearance of {x,y}, k in the following table means that (x,y) = X_k:

  {-2 (a^2 b+a b^2+a^2 c+b^2 c+a c^2+b c^2), -2 (a b+a c+b c), 21385

  {-a b c, -((2 a b c)/(a+b+c))}, 1459

  {-a^2 b-a b^2-a^2 c-b^2 c-a c^2-b c^2, -2 (a b+a c+b c)}, 17494

  {-a^3-b^3-c^3, -a^2-b^2-c^2}, 47729

  [and others].

• X₄₇₃₂₀–X₄₈₃₄₀ — Points in a [X₁X_513, X₁X_514] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁: b c (2a - b - c) α + c a (2b - c - a) β + a b (2 c - a - b) γ = 0.

  L₂: (b^2 + c^2 - a b - a c) α (c^2 + a^2 - b c - b a) β (a^2 + b^2 - c a - c b) γ = 0.

  The origin is given by (0,0) = X₁ =\(a : b : c\).

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) (a (a - b)(a - c) - a x - y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 2, and y is symmetric and homogeneous of degree 3.

  The appearance of {x,y}, k in the following table means that (x,y) = X_k:

  {-2 (a b+a c+b c), a b c}, 48144}

  {-2 (a b+a c+b c), a^3+b^3+c^3}, 47676

  {-((2 a b c)/(a+b+c)), a b c}, 1459

  {-2 (a b+a c+b c), 2 a b c}, 1019

  {-2 (a b+a c+b c), 2 (a+b+c) (a b+a c+b c)}, 47683

  {-((2 a b c)/(a+b+c)), 2 a b c}, 3737

  [and others].

• X₄₇₃₈₂–X₄₈₃₉₁ — Points in a [X₂X_513, X₂X_523] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = the line X₃X₅₁₃ with coefficients given by the barycentrics for the isotomic conjugate of X₂₉₉₀, shown at X₄₈₃₈₀

  L₂ = the line X₃X₅₁₄ with coefficients given by the barycentrics for the isotomic conjugate of X₂₉₈₉, shown at X₄₈₃₈₁

  The origin is given by (0,0) = X₃, the circumcenter.

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) (a^2(a - b)(a - c)(a^2 - b^2 - c^2) + a x - y]) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 5, and y is symmetric and homogeneous of degree 6.

  The appearance of {x,y}, k in the following list means that (x,y) = X_k:

  {-((2 a^2 b^2 c^2)/(a+b+c)), -a^2 b^2 c^2}, 39226

  {0,-2 a^2 b^2 c^2}, 44408

  {0,-a^2 b^2 c^2}, 39476

  {0,0},3}

  {(-2*a^2*b^2*c^2)/(a + b + c), -2*a^2*b^2*c^2}, 48382

  [and others].

• X₄₇₃₉₂–X₄₈₄₁₀ — Points in a [L(31),L(32)] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = the line L(32) = X₃₂₅X₅₂₃ = [a^2, b^2, c^2];

  L₂ = the line L(31) = X₅₁₄X₆₆₁ = [a,b,c].

  The origin is given by (0,0) = X₆₉₃ = b c (b + c) : : .

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) (b c - (b+c) x + y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 1, and y is symmetric and homogeneous of degree 2.

  Note that the (L(31),L(32)) coordinate system is not the same as the (L(32),L(31) system; see the preamble just before X₄₇₆₅₀.,

  The appearance of {x,y}, k in the following table means that (x,y) = X_k:

  {-2 (a+b+c), -2 (a b+a c+b c)}, 47665)

  {-((2 (a^2+b^2+c^2))/(a+b+c)), -2 (a^2+b^2+c^2)), 47685)

  {-((2 (a^2+b^2+c^2))/(a+b+c)), -((2 a b c)/(a+b+c))}, 47706)

  {-((2 (a b+a c+b c))/(a+b+c)), -2 (a b+a c+b c)}, 48080)

  {-2 (a+b+c), -a b-a c-b c}, 4838)

  [and others].

• X₄₇₄₁₂–X₄₈₄₃₈ — Odd minor triangle centers

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ asA triangle center P =\(p(a,b,c) : q(a,b,c) : r(a,b,c)\) is even if p(a,b,c) = p(a,c,b) and odd if p(a,b,c) = - p(a,c,b); or equivalently, p(a,b,c) = (b - c) u(a,b,c), where \(u(a,b,c) : u(b,c,a) : u(c,a,b)\) is even. These definitions follow C. Kimberling, “Functional equations associated with triangle geometry,” Aequationes Mathematicae 45 (1993) 127-152. The triangle center P is minor if p(a,b,c) is invariant of a. (This definition is not closely related to “major” triangle center, defined elsewhere as a center that can be expressed in barycentrics \(m(A,B,C) : m(B,C,A) : m(C,A,B)\) such that m(A, B, C) is invariant of B and C.)

  The odd minor centers in this section are all polynomial centers of degree 3, with first barycentric given by the form (b - c)(h*(b^2 + c^2) + k* b c), where h and k are real numbers, not both zero.

  The appearance of {h,k,i} in the following list means that X_i = (b - c)(h*(b^2 + c^2) + k* b c):

  {0,1}, 693}

  {1,-2}, 6545}

  {1,-1}, 3776}

  {1,0}, 16892}

  {1,1}, 824}

  [and others].

• X₄₇₄₅₄–X₄₈₅₄₂ — Centers related to anti-Auriga triangles

  contributed by César Eliud Lozada, May 12, 2022. The 1st- and 2nd- anti-Auriga triangles were introduced in the preamble just before X₄₅₃₄₅.

• X₄₇₅₄₃–X₄₈₅₈₀ — Points in a [X₂X_513, X₂X_514] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁: (2bc-ab-ac) α + (2ca-bc-ba) β + (2ab-ca-cb) γ = 0.

  L₂: (2a-b-c) α + (2b-c-a) β + (2c-a-b) γ = 0.

  The origin is given by (0,0) = X₂ =\(1 : 1 : 1\) .

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) ((a-b)(a-c) + ax + y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 1, and y is symmetric and homogeneous of degree 2.

  For the “reversed” coordinate system, [L₂,L₁], see X₄₇₇₅₄ - X₄₇₈₄₅.

  The appearance of {x,y},k in the following list means that (x,y) = X_k:

  {-2 (a+b+c), -2 (a b+a c+b c)}, 47774

  {-((2 (a b+a c+b c))/(a+b+c)), -a b-a c-b c}, 47826

  {-2 (a+b+c), 0}, 47759

  {-((2 (a^2+b^2+c^2))/(a+b+c)), 0}, 48164

  {-((2 (a b+a c+b c))/(a+b+c)), 0}, 47821

  {-2 (a+b+c), 1/2 (a^2+b^2+c^2)}, 47764

  [and others].

• X₄₇₅₈₂–X₄₈₆₂₆ — Points in the [ [bc,ca,ab], [a,b,c] ] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁: bc α + ca β + ab γ = 0.

  L₂: a α + b β + c γ = 0.

  The origin is given by (0,0) = X₆₆₁ =\(a(b^{2}-c^{2}) : b(c^{2}-a^{2}) :c(a^{2}-b^{2})\).

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (b - c) (ab + ac - a x - y) : : ,

  where, as functions of a,b,c, the coordinate x is symmetric and homogeneous of degree 1, and y is symmetric and homogeneous of degree 2.

  For the “reversed” coordinate system, [L₂,L₁], see X₄₇₉₀₀ - X₄₈₁₅₁.

  The appearance of {x,y},k in the following list means that (x,y) = X_k and that k ≥ 48582.

  {-2*(a + b + c), (-2*a*b*c)/(a + b + c)}, 48582

  {(-2*(a^2 + b^2 + c^2))/(a + b + c), -2*(a*b + a*c + b*c)}, 48583

  {-2*(a + b + c), -((a*b*c)/(a + b + c))}, 48584

  {(-2*(a^2 + b^2 + c^2))/(a + b + c), -a^2 - b^2 - c^2}, 48585

  [and others].

• X₄₇₆₂₉–X₄₈₆₄₀ — 2nd degree even minor triangle centers

  contributed by Clark Kimberling and Peter Moses, undated. Centers in this section are of the form h*(b^2 + c^2) + k* b c, where h and k are real numbers, not both 0. All such points lie on the line X₇₅X₁₄₁.

  The appearance of {h,k},i in the following list means that h*(b^2 + c^2) + k* b c = X_i.

  {0,1}, 75

  {1,-4}, 7263

  {1,-2}, 1086

  {1,-1}, 3662

  {1,0}, 141

  {1,2}, 594

  {1,4}, 4665

  {2,-1}, 17227

  [and others].

• X₄₇₆₄₁–X₄₈₆₅₄ — 3rd degree even minor triangle centers

  contributed by Clark Kimberling and Peter Moses, undated. Centers in this section are of the form h*(b^3 + c^3) + k* b c (b + c), where h and k are real numbers, not both 0. All such points lie on the line X₃₂₁X₂₈₈₇₁.

  The appearance of {h,k},i in the following list means that h*(b^3 + c^3) + k* b c (b + c) = X_i.

  {1,-1}, 3120

  {1,0}, 2887}

  {1,2}, 3773}

  {1,3}, 6535}

  {1, -4}, 48641

  [and others].

• X₄₇₆₅₅–X₄₈₆₈₃ — Centers related to anti-Ehrmann-mid triangle

  :contributed by César Eliud Lozada, May 13, 2022. The anti-Ehrmann-mid triangle is introduced in the preamble just before X₄₅₃₄₅..

• X₄₇₆₈₄–X₄₈₇₂₀ — Centers related to anti-inner-Garcia triangle

  contributed by César Eliud Lozada, May 13, 2022. The anti-inner-Garcia triangle was introduced in the preamble just before X₄₅₃₄₅.

• X₄₇₇₂₂–X₄₈₇₉₃ — Centers related to anti-Kenmotu-free-vertices triangles

  contributed by César Eliud Lozada, May 14, 2022. The anti-Kenmotu-free-vertices triangles were introduced in the preamble just before X₄₅₃₄₅.

• X₄₇₇₉₈–X₄₈₈₃₃ — Points in the [ [b-c,c-a,a-b], [(b^2 - c^2)(a^2 - b^2 - c^2), (c^2 - a^2)(b^2 - c^2 - a^2), (a^2 - b^2)(c^2 - a^2 - b^2)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = Nagel line: (b-c) α + (c-a) β + (a-b) γ = 0.

  L₂ = Euler line: (b^2 - c^2)(a^2 - b^2 - c^2) α + (c^2 - a^2)(b^2 - c^2 - a^2) β + (a^2 - b^2)(c^2 - a^2 - b^2) γ = 0.

  The origin is given by (0,0) = X₂ =\(1 : 1 : 1\).

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  \(u : v : w\) = (b-c)(a-b)(a-c) (a+b+c)^2 + (2a-b-c) x + (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) y : : ,

  where, as functions of a,b,c, the coordinate x is antisymmetric and homogeneous of degree 1, and y is antisymmetric and homogeneous of degree 2n; viz., x and y are of the form (b-c)(a-b)(a-c)z(a,b,c), where z(a,b,c) is symmetric in a,b,c..

  The appearance of {x,y},k in the following list means that (x,y) = X_k.

  {-2 (a-b) (a-c) (b-c) (a+b+c), 0}, 8

  {-((a-b) (a-c) (b-c) (a+b+c)), 0}, 3679

  {-(1/2) (a-b) (a-c) (b-c) (a+b+c), 0}, 10

  {0, -(((a-b) (a-c) (b-c))/(a^2+b^2+c^2))}, 11359

  {0, 0}, 2

  [and others].

• X₄₈₈₃₄–X₄₈₈₇₀ — Points in a [Euler line, Nagel line] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁,L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = Euler line: (b^2 - c^2)(a^2 - b^2 - c^2) α + (c^2 - a^2)(b^2 - c^2 - a^2) β + (a^2 - b^2)(c^2 - a^2 - b^2) γ = 0.

  L₂= Nagel line: (b-c) α + (c-a) β + (a-b) γ = 0.

  The origin is given by (0,0) = X₂ =\(1 : 1 : 1\).

  Barycentrics \(u : v : w\) for a point U = (x,y) in this system are given by

  \(u : v : w\) = (a - b) (a - c) (b - c) (a + b + c)^2 + (2 a^4 - a^2 b^2 - b^4 - a^2 c^2 + 2 b^2 c^2 - c^4) x + (2 a - b - c) y,

  where, as functions of a,b,c, the coordinate x is antisymmetric and homogeneous of degree 1, and y is antisymmetric and homogeneous of degree 2. For a [Nagel line, Euler line] coordinate system, see trhe preamble just before X₄₈₇₉₈.

  The appearance of {x,y},k in the following list means that (x,y) = X_k.

  {-((2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2)), -2 (a-b) (a-c) (b-c) (a+b+c)}, 48798

  {-((2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2)), -((a-b) (a-c) (b-c) (a+b+c))}, 48807

  {-((2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2)), 0}, 48813

  [and others].

• X₄₈₈₇₂–X₄₈₉₁₀ — Points in a [Brocard axis, Euler line] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = Brocard axis: b^2 c^2 (b^2 - c^2) α + c^2 a^2 (c^2 - a^2) β + a^2 b^2 (a^2 - b^2) γ = 0.

  L₂ = Euler line: (b^2 - c^2)(a^2 - b^2 - c^2) α + (c^2 - a^2)(b^2 - c^2 - a^2) β + (a^2 - b^2)(c^2 - a^2 - b^2) γ = 0.

  The origin is given by (0, 0) = X₃ = a^2(a^2-b^2-c^2) : : .

  Barycentrics \(u : v : w\) for a point U = (x, y) in this system are given by

  \(u : v : w\) = a^2(a^2-b^2-c^2)(b^2-c^2)(c^2-a^2)(a^2-b^2) - a^2(a^2(b^2+c^2) - b^4 - c^4) x + (a^2(-2a^2+b^2+c^2) + (b^2-c^2)^2)y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric and homogeneous of degree 4, and y is antisymmetric and homogeneous of degree 6.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c) (a+b+c), -2 a (a-b) b (a-c) (b-c) c}, 12702

  {-((2 (a^2-b^2) (a^2-c^2) (b^2-c^2))/(a^2+b^2+c^2)), 0}, 1350

  {-((2 (a^2-b^2) (a^2-c^2) (b^2-c^2))/(a^2+b^2+c^2)), (a^2-b^2) (a^2-c^2) (b^2-c^2)}, 1352

  [and othersj].

• X₄₈₉₁₅–X₄₈₉₄₄ — Points in a [Euler line, Brocard axis] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ = Euler line: (b^2 - c^2)(a^2 - b^2 - c^2) α + (c^2 - a^2)(b^2 - c^2 - a^2) β + (a^2 - b^2)(c^2 - a^2 - b^2) γ = 0.

  L₂ = Brocard axis: b^2 c^2 (b^2 - c^2) α + c^2 a^2 (c^2 - a^2) β + a^2 b^2 (a^2 - b^2) γ = 0.

  The origin is given by (0, 0) = X₃ = a^2(a^2-b^2-c^2) : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = a^2(a^2-b^2-c^2)(b^2-c^2)(c^2-a^2)(a^2-b^2) + (a^2(-2a^2+b^2+c^2) + (b^2-c^2)^2)x - a^2(a^2(b^2+c^2) - b^4 - c^4) y : : ,

  where, as polynomials in a, b, c, the coordinate x is antisymmetric and homogeneous of degree 6, and y is antisymmetric and homogeneous of degree 4.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 a (a-b) b (a-c) (b-c) c, -2 (a-b) (a-c) (b-c) (a+b+c)}, 12702}

  {-2 (a^2-b^2) (a^2-c^2) (b^2-c^2), -((2 (a^2-b^2) (a^2-c^2) (b^2-c^2))/(a^2+b^2+c^2))}, 48872

  {-2 (a^2-b^2) (a^2-c^2) (b^2-c^2), -(((a^2-b^2) (a^2-c^2) (b^2-c^2))/(a^2+b^2+c^2))}, 48879

  {-2 (a^2-b^2) (a^2-c^2) (b^2-c^2), 0}, 1657

  {-2 (a^2-b^2) (a^2-c^2) (b^2-c^2), ((a^2-b^2) (a^2-c^2) (b^2-c^2))/(a^2+b^2+c^2)}, 48896

  [and others]

• X₄₈₉₄₇–X₄₈₉₇₉ — Centers related to 1st anti-Parry triangle

  contributed by César Eliud Lozada, May 18, 2022. The 1st anti-Parry triangle was introduced in the preamble just before X₄₅₃₄₅.

• X₄₈₉₈₀–X₄₉₀₁₁ — Centers related to 2nd anti-Parry triangle

  contributed by César Eliud Lozada, May 18, 2022. The 2nd anti-Parry triangle was introduced in the preamble just before X₄₅₃₄₅.

• X₄₉₀₁₂–X₄₉₁₀₁ — Centers related to 3rd- and 4th- anti-tri-squares-central triangles

  contributed by César Eliud Lozada, May 18, 2022. The 3rd- and 4th- anti-tri-squares-central triangles were introduced in the preamble just before X₄₅₃₄₅.

• X₄₉₁₀₂–X₄₉₁₂₀ — Centers related to anti-X3-ABC reflections triangle

  contributed by César Eliud Lozada, May 18, 2022. The anti-X3-ABC reflections triangle was introduced in the preamble just before X₄₅₃₄₅.

• X₄₉₁₂₇–X₄₉₁₄₀ — Combos (1+h)*X₃ - h*X₄ on the Euler line

  contributed by Clark Kimberlikng and Peter Moses, undated. If h is a constant or other function symmetric in a,b,c and of degree 0 of homogeneity, then (1+h)*X₃-h*X₄, like every combo of points on the Euler line, is also on the Euler line. The combo (1+h)*X₃-h*X₄ can also be expressed as X₃ + h*X₃₀.

  In the following list, the appearance of h,k means that (1+h)*X₃-h*X₄ = X_k.

  -10, 49133

  -9,11541

  -8, 49134

  -7, 49135

  -6, 49136

  -5,33703

  -4,5073

  -3,3146

  -2,382

  -1,4

  -2/3,381

  -1/2,5

  -1/3,2

  [and others].

• X₄₉₁₄₃–X₄₉₂₀₇ — Centers related to anti-inner-Yff and anti-outer-Yff triangles

  contributed by César Eliud Lozada, May 20, 2022. The anti-inner-Yff and anti-outer-Yff triangles were introduced in the preamble just before X₄₅₃₄₅.

• X₄₉₂₀₈–X₄₉₂₉₁ — Centers related to 1st- and 2nd- Kenmotu-centers triangles

  contributed by César Eliud Lozada, May 20, 2022. The 1st- and 2nd- Kenmotu-centers triangles were introduced in the preamble just before X₄₄₅₈₂.

• X₄₉₂₇₂–X₄₉₃₀₃ — Points in a [[a,b,c], [a(b+c), b(c+a), c(a+b)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: a α + b β + c γ = 0.

  L₂ is the line a(b+c) α + b(c+a) β + c(a+b) γ = 0.

  The origin is given by (0, 0) = X₆₉₃ =\(bc(b-c) : ca(c-a) : ab(a-b)\) .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = (b-c)(-bc + x - ay) : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric and homogeneous of degree 2, and y is antisymmetric and homogeneous of degree 1.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a^2+b^2+c^2), -((2 (a^2+b^2+c^2))/(a+b+c))}, 47689}

  {-2 (a^2+b^2+c^2), -((a^2+b^2+c^2)/(a+b+c))}, 47693

  {-2 (a b+a c+b c), -a-b-c}, 26824

  {-2 (a^2+b^2+c^2), 0}, 47662

  [and others].

• X₄₉₃₀₅–X₄₉₃₇₂ — Centers related to 1st- and 2nd- anti-Kenmotu-centers triangles triangles

  contributed by César Eliud Lozada, May 21, 2022. The 1st- and 2nd- anti-Kenmotu-centers triangles triangles were introduced in the preamble just before X₄₅₃₄₅.

• X₄₉₃₇₃–X₄₉₄₄₄ — Centers related to anti-Lucas(±1)-homothetic triangles

  contributed by César Eliud Lozada, May 21, 2022. The anti-Lucas(±1)-homothetic triangles were introduced in the preamble just before X₄₅₃₄₅.

• X₄₉₄₄₅–X₄₉₅₀₀ — Points in a [[bc(b-c),ca(c-a),ab(a-b)], [a(b^2-c^2),b(c^2-a^2),c(a^2-b^2)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: bc(b-c) α + ca(c-a) β + ab(a-b) γ = 0.

  L₂ is the line a(b^2-c^2) α + b(c^2-a^2) β + c(a^2-b^2) γ = 0.

  The origin is given by (0, 0) = X₁ =\(a : b : c\) .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -a(a+b)(a+c)(b-c)(ab+ac+bc) - a(ab+ac-b^2-c^2)x + (b+c)(a^2-bc)y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric and homogeneous of degree 3, and y is antisymmetric and homogeneous of degree 3.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-((2 (a-b) (a-c) (b-c) (a b+a c+b c))/(a^2+b^2+c^2)), 0}, 16496

  {-2 (a-b) (a-c) (b-c), 2 (a-b) (a-c) (b-c)}, 3632

  {-((a-b) (a-c) (b-c)), -((a-b) (a-c) (b-c))}, 192

  {-((a (a-b) b (a-c) (b-c) c)/(a^3+b^3+c^3)), -(((a-b) (a+b) (a-c) (b-c) (a+c) (b+c))/(a^3+b^3+c^3))}, 3891

  {-((a-b) (a-c) (b-c)), 0}, 984

  [and others].

• X₄₉₅₀₁–X₄₉₅₃₆ — Points in a [[bc(b-c),ca(c-a),ab(a-b)], [a^2(b-c),b^2(c-a), c^2(a-b))]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: bc(b-c) α + ca(c-a) β + ab(a-b) γ = 0.

  L₂ is the line a^2(b-c) α + b^2(c-a) β + c^2(a-b) γ = 0.

  The origin is given by (0, 0) = X₉₈₄ =\(a(b-c) : b(c-a) : c(a-b)\) .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -a(a+b)(a+c)(b-c)(b^2+bc+c^2) - a(ab+ac-b^2-c^2)x + (ab^2+ac^2-b^2c-bc^2)y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric and homogeneous of degree 3, and y is antisymmetric and homogeneous of degree 3.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c), (a-b) (a-c) (b-c)}, 49450

  {-2 (a-b) (a-c) (b-c), 2 (a-b) (a-c) (b-c)}, 49459

  {-((a-b) (a-c) (b-c)), -((a-b) (a-c) (b-c))}, 31302

  {-((a-b) (a-c) (b-c)), 0}, 49448

  {-(((a-b) (a-c) (b-c) (a^2+b^2+c^2))/(a b+a c+b c)), ((a-b) (a-c) (b-c) (a^2+b^2+c^2))/(2 (a b+a c+b c))}, 3883

  {-((a-b) (a-c) (b-c)), (a-b) (a-c) (b-c)}, 8

  [and others].

• X₄₉₅₃₇–X₄₉₆₃₃ — Centers related to 1st- and 2nd- Savin triangles

  contributed by César Eliud Lozada, May 22, 2022. 1st- and 2nd- Savin triangles were introduced in the preamble just before X₄₄₃₀₁.

• X₄₉₆₃₄–X₄₉₆₆₈ — Centers related to Jenkins triangles

  contributed by César Eliud Lozada, May 22, 2022.

  1st- and 2nd- Jenkin triangles were introduced in “Hechos Geométricos en el Triángulo”, by Angel Montesdeoca. These two triangles, and three other related Jenkins triangles are defined in the Index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html).

• X₄₉₆₇₅–X₄₉₇₁₅ — Points in a [[bc(b-c),ca(c-a),ab(a-b), [(b-c)^3, (c-a)^3,(a-b)^3]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  p> L₁ is the line: bc(b-c) alpha + ca(v-a) β ab(a-b) γ = 0.

  L₂ is the line (b-c)^3 α + (c-a)^3 β + (a-b)^3 γ = 0.

  The origin is given by (0, 0) = X₂₃₈ = a^2 - b c : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -a(a-b)(a-c)(b-c)(a^2 - b c) - a (a b + a c - b^2 - c^2) x + (2a - b - c)(a^2 + b^2 + c^2 - a b - a c - b c) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 3, and y is antisymmetric of degree 3.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c), (a-b) (a-c) (b-c)}, 32850

  {-((a-b) (a-c) (b-c)), -2 (a-b) (a-c) (b-c)}, 3633

  {-((a-b) (a-c) (b-c)), -((2 a (a-b) b (a-c) (b-c) c)/(a^3+b^3+c^3))}, 49494

  {-((a-b) (a-c) (b-c)), -((2 (a-b) (a-c) (b-c) (a b+a c+b c))/(a^2+b^2+c^2))}, 49495

  {-((a-b) (a-c) (b-c)), -((a-b) (a-c) (b-c))}, 145

  [and others]

• X₄₉₇₁₆–X₄₉₇₄₉ — Points in a [[b^2 - c^2, c^2 - a^2, a^2 - b^2], [(b^2 - c^2)(a^2 - b^2 - c^2), (c^2 - a^2)(b^2 - c^2 - a^2), (a^2 - b^2)(c^2 - a^2 - b^2) ]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. L₁ is the line: (b^2 - c^2) α + (c^2 - a^2) β (a^2 - b^2) γ = 0.

  L₂ is the line (b^2 - c^2)(a^2 - b^2 - c^2) α + (c^2 - a^2)(b^2 - c^2 - a^2) β + (a^2 - b^2)(c^2 - a^2 - b^2) γ = 0 (Euler line).

  The origin is given by (0,0) = X₂ =\(1 : 1 : 1\) .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -2(a^2 - b^2)(a^2 - c^2)(b^2 - c^2) + (-2a^2 + b^2 + c^2) x + (-2a^4 + b^4 + c^4 + a^2 b^2 + a^2 c^2 - 2 b^2 c^2 ) y : : ,

  where, as functions of a, b, c, the coordinate x is symmetric of degree 4, and y is antisymmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c) (a+b+c), 0}, 3578

  {-((2 (a-b) (a+b) (a-c) (b-c) (a+c) (b+c))/(a^2+b^2+c^2)), 0}, 599

  {-((2 (a-b) (a+b) (a-c) (b-c) (a+c) (b+c))/(a b+a c+b c)), 0}, 17346

  {-2 (a-b) (a-c) (b-c) (a+b+c), (2 (a-b) (a-c) (b-c))/(a+b+c)}, 8

  {-(((a-b) (a+b) (a-c) (b-c) (a+c) (b+c))/(a^2+b^2+c^2)), 0}, 141

  [and others].

• X₄₉₇₅₀–X₄₉₇₈₃ — Points in a [[a,b,c], [b-c,c-a,a-b]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: a α + b β c γ = 0.

  L₂ is the line (b-c) α + (c-a) β + (a-b) γ = 0 (Nagel line).

  The origin is given by (0,0) = X₃₉₁₂ = b^2 + c^2 - a b - a c : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = b^2 + c^2 - a(b + c) + (b-c) x + (2a - b - c) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 1, and y is symmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-((2 (a-b) (a-c) (b-c))/(a b+a c+b c)), (2 a b c)/(a b+a c+b c)}, 41773

  {-(((a-b) (a-c) (b-c))/(a^2+b^2+c^2)), (a^3+b^3+c^3)/(a^2+b^2+c^2)}, 24281

  {0, -a-b-c}, 6542

  {0, -((a^2+b^2+c^2)/(a+b+c))}, 32847

  {0, 0}, 3912

  {0, 1/2 (a+b+c)}, 3008

  {0, a+b+c}, 239

  {0, (a^2+b^2+c^2)/(a+b+c)}, 1

  {0, (a b+a c+b c)/(a+b+c)}, 10

  [and others].

• X₄₉₇₉₄–X₄₉₉₇₈ — Centers related to Fermat-Dao-Nhi triangles

  contributed by César Eliud Lozada, May 26, 2022. Fermat-Dao-Nhi triangles were introduced in the preamble just before X₃₃₆₀₂.

• X₄₉₉₇₉–X₅₀₀₀₃ — Points in a [[bc,ca,ab], [b-c,c-a,a-b]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: bc α + ca β ab γ = 0.

  L₂ is the line (b-c) α + (c-a) β + (a-b) γ = 0 (Nagel line).

  The origin is given by (0,0) = X₈₉₉ = a(2bc-ab-ac) : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -a(a b + a c - 2 b c) - a(b-c) x + (2a - b - c) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 1, and y is symmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-(((a-b) (a-c) (b-c) (a+b+c))/(2 a b c)), (a^3+b^3+c^3)/(a+b+c)}, 72

  {0, -a b-a c-b c}, 19998

  {0, -((a b c)/(a+b+c))}, 31855

  {0, 0}, 899

  {0, 1/2 (a b+a c+b c)}, 4871

  {0, a b+a c+b c}, 29824

  {0, (2 a b c)/(a+b+c)}, 1149

  {-1/2*((a - b)*(a - c)*(b - c)*(a + b + c))/(a*b*c), (a + b + c)^2/2}, 49979

  [and others].

Part 26

• X₅₀₀₁₀–X₅₀₀₃₀ — Points in a [[b+c,c+a,a+b], [b-c,c-a,a-b]] coordinate system

  contibuted by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: (b+c) α + (c+a) β (a+)b γ = 0.

  L₂ is the line (b-c) α + (c-a) β + (a-b) γ = 0 (Nagel line).

  The origin is given by (0,0) = X₂₃₉ = a^2 - bc : b^2 - ca : c^2 - ab .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -2(a^2 - bc) - (b-c)x + (2a - b - c) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 1, and y is symmetric of degree 1.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-((2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2)), (2 (a^3+b^3+c^3))/(a^2+b^2+c^2)}, 69

  {-(((a-b) (a-c) (b-c))/(a b+a c+b c)), ((a+b) (a+c) (b+c))/(a b+a c+b c)}, 49755

  {-(((a-b) (a-c) (b-c) (a+b+c))/(2 a b c)), 1/2 (a+b+c)}, 49759

  {-(((a-b) (a-c) (b-c) (a+b+c))/(2 (a+b) (a+c) (b+c))), 1/2 (a+b+c)}, 49760

  {0, -2 (a+b+c)}, 20016

  {0, -a-b-c}, 49770

  {0, 0}, 239

  [and others].

• X₅₀₀₃₂–X₅₀₀₄₀ — Miyamoto-Lozada centers

  . The centers X₅₀₀₃₂ - X₅₀₀₄₀, except for X₅₀₀₃₉, were noted and conjectured by Keita Miyamoto, and confirmed by César Lozada. All the barycentric coordinates and other properties were found by Lozada, who also discovered X₅₀₀₃₉.

  These centers are associated with circles tangent to the 3 excircles. In particular, X₅₀₀₃₇, X₅₀₀₃₈, X₅₀₀₄₀ involve Hart circles of the excircles. (See Mathworld: Hart’s Theorem.)

  Here, the notation (O) means a circle with center O. Let (E_a), (E_b), (E_c) denote the excircles, and let (S) be their radical circle. There are 8 lines or circles each of which is tangent to (E_a), (E_b), (E_c) simultaneously. These 8 are:

  sidelines AB, BC, CA;

  nine-point circle (N);

  Apollonius circle (Ap); note that (Ap) is the (S)-inverse of (N);

  The three Jenkins circles (J_a), (J_b), (J_c); the A-Jenkins circle (J_a) is the (S)–inverse of BC; and (J_b) and (J_c) are defined cyclically.

  Hart’s theorem states that generally, other than (E_a), (E_b), (E_c), there are 14 circles in total each of which is tangent to 4 of the 8 simultaneously. Following are the 14 Hart circles of the excircles:

  The incircle (I) is tangent to AB, BC, CA, internally tangent to (N).

  The Moses hull circle (M) is (internally or externally) tangent to (J_a), (J_b), (J_c), (Ap). (If S lies on (I), then (M) is a line.)

  (O₁) is tangent to AB and CA, internally tangent to (J_b) and (J_c); the circles (O₂) and (O₃) are defined cyclically.

  (O₄) is tangent to BC, internally tangent to (Ap) and (J_a), and externally tangent to (N); the circles (O₅) and (O₆) are defined cyclically.

  (O₇) is tangent to BC, internally tangent to (N), and externally tangent to (J_b) and (J_c); the circles (O₈) and (O₉) are defined cyclically.

  (O10) is tangent to AB and CA, and externally tangent to (Ap) and (J_a); the circles (O11) and (O12) are defined cyclically.

  Among the 14 Hart circles, the 6 Hart circles (O₁), (O₂), (O₃), (O₄), (O₅), (O₆) are orthogonal to (S), while the 4 Hart circles (M), (O10), (O11), (O12) are the (S)-inverses of the other 4: (I), (O₇), (O₈), (O₉), respectively. Clearly, IM, O7O10, O8O11, O9O12 concur in S = X₁₀. Also, O₁O₄, O₂O₅, O₃O₆ concur in X₅₀₀₃₇.

• X₅₀₀₄₁–X₅₀₀₇₃ — Points in a [[a(b-c),b(c-a),c(a-b)], [(b^2-c^2)(a^2-b^2-c^2), (c^2-a^2)(b^2-c^2-a^2), (a^2-b^2)(a^2 - b^2 - c^2)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: a(b+c) α + b(c+a) β (a+)b γ = 0.

  L₂ is the line (b^2-c^2)(a^2-b^2-c^2) α + (c^2-a^2)(b^2-c^2-a^2) β + (a^2-b^2)(a^2 - b^2 - c^2) γ = 0 (Euler line).

  The origin is given by (0,0) = X₂ = 1 1 : 1 .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -(a-b)(a-c)(b-c)(a^3+b^3+c^3+a^2b+ab^2+b^2c+bc^2+c^2a+a^2c+2abc) - (ab + ac - 2bc)x - (2a^4-b^4-c^4+2b^2c^2-a^2b^2-a^2c^2) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 4, and y is antisymmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c) (a+b+c), -((2 (a-b) (a-c) (b-c))/(a+b+c))}, 8

  {-((a-b) (a-c) (b-c) (a+b+c)), -(((a-b) (a-c) (b-c))/(a+b+c))}, 3679

  {-(1/2) (a-b) (a-c) (b-c) (a+b+c), -(((a-b) (a-c) (b-c))/(2 (a+b+c)))}, 10

  [and others].

• X₅₀₀₇₄–X₅₀₁₃₃ — Points in a [[b-c,c-a,a-b],[a(b-c),b(c-a),c(a-b)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: (b-c) α + (c-a) β (a-b) γ = 0.

  L₂ is the line a(b-c) α + b(c-a) β c(a-b) γ = 0.

  The origin is given by (0,0) = X₂ = 1 1 : 1 .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -(a-b)(a-c)(b-c) + (-2 a + b + c) x + (a b + a c - 2 b c) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 2, and y is antisymmetric of degree 1.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-((2 (a-b) (a-c) (b-c) (a+b+c))/(a^2+b^2+c^2)), -((2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2))}, 69

  {-((2 (a-b) (a-c) (b-c) (a+b+c))/(a^2+b^2+c^2)), -(((a-b) (a-c) (b-c))/(2 (a^2+b^2+c^2)))}, 17372

  {-((2 (a-b) (a-c) (b-c))/(a+b+c)), 0}, 8

  [and others].

• X₅₀₁₄₄–X₅₀₁₅₀ — Points in a [[2a^2-b^2-c^2,2b^2-c^2-a^2,2c^2-a^2-b^2], [L2 = [(b^2-c^2)(a^2-b^2-c^2), (c^2- a^2)(b^2-c^2-a^2), (a^2- b^2)(a^2-b^2-c^2)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: (2a^2-b^2-c^2) α + (c^2- a^2)(b^2-c^2-a^2) β + 2c^2-a^2-b^2 γ = 0.

  L₂ is the line (b^2-c^2)(a^2-b^2-c^2) α + (c^2- a^2)(b^2-c^2-a^2) β + (a^2- b^2)(a^2-b^2-c^2) γ = 0 (Euler line).

  The origin is given by (0,0) = X₂ =\(1:1:1\) = G : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = -2(a^6 - a^4(b^2+c^2) - a^2 (b^4 - 3 b^2 c^2 + c^4) + (b^2 - c^2)(b^4 - c^4)) + 3(b^2-c^2) x - (a^2 (2a^2 - b^2 - c^2) - (b^2 - c^2)^2) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 4, and y is symmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c) (a+b+c), 1/2 (a^2+b^2+c^2)}, 16304

  {-((2 (a^2-b^2) (a^2-c^2) (b^2-c^2))/(a^2+b^2+c^2)), 1/2 (a^2+b^2+c^2)}, 16334

  {-((a-b) (a-c) (b-c) (a+b+c)), 1/2 (a^2+b^2+c^2)}, 16309} {-(((a^2-b^2) (a^2-c^2) (b^2-c^2))/(a^2+b^2+c^2)), 1/2 (a^2+b^2+c^2), 16321

  {-(1/2) (a-b) (a-c) (b-c) (a+b+c), 1/2 (a^2+b^2+c^2)}, 16305

  [and others].

• X₅₀₁₉₉–X₅₀₂₀₈ — Points on the Euler line

  contributed by Clark Kimberling and Peter Moses, undated.In the plane of a triangle ABC, let

  P = point on Nagel line;

  D = point not on Nagel line or Euler line;

  U = point on Nagel line, other than U and G;

  L = line through U parallel to PD;

  U' = L^(Euler line).

  For centers X₅₀₁₉₉-X₅₀₂₀₈, we take P = X₁ and D = X₆. The appearance of (i,j) in the following list means that if if U = X_i then U' = X_j: (43,50199), (239,50200), (306,50201), (551, 50202), (946, 50203), (997, 50204), (1125,50205), (1210), 50206), (1698,50207), (1737,50208).

• X₅₀₂₁₅–X₅₀₂₃₆ — Points in a [[a^2 (b^2 - c^2), b^2 (c^2 - a^2), c^2 (a^2 - b^2)], [L2 = [(b^2-c^2)(a^2-b^2-c^2), (c^2- a^2)(b^2-c^2-a^2), (a^2- b^2)(a^2-b^2-c^2)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: a^2 (b^2 - c^2) α + b^2 (c^2 - a^2) β + c^2 (a^2 - b^2) γ = 0.

  L₂ is the line (b^2-c^2)(a^2-b^2-c^2) α + (c^2- a^2)(b^2-c^2-a^2) β + (a^2- b^2)(a^2-b^2-c^2) γ = 0 (Euler line).

  The origin is given by (0,0) = X₂ =\(1:1:1\) = G .

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = - (a^2 - b^2)(a^2 - c^2)(b^2 - c^2)(a^2 + b^2 + c^2) + (2 b^2 c^2 - a^2 b^2 - a^2 c^2) x + (b^4 + c^4 - 2a^4 + a^2 b^2 + a^2 c^2 - 2 b^2 c^2) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 4, and y is antisymmetric of degree 4.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c) (a+b+c), -((2 (a-b) (a-c) (b-c) (a b+a c+b c))/(a+b+c))}, 8

  {-2 (a-b) (a-c) (b-c) (a+b+c), -((a-b) (a-c) (b-c) (a+b+c))}, 3578

  {-((2 (a-b) (a-c) (b-c) (a b+a c+b c))/(a+b+c)), -(((a-b) (a-c) (b-c) (a b+a c+b c))/(a+b+c))}, 49717

  {-((a-b) (a-c) (b-c) (a+b+c)), -(((a-b) (a-c) (b-c) (a b+a c+b c))/(a+b+c))}, 3679

  {-((a-b) (a-c) (b-c) (a+b+c)), -(1/2) (a-b) (a-c) (b-c) (a+b+c)}, 49724

  [and others].

• X₅₀₂₃₇–X₅₀₂₄₄ — Points on the Euler line

  contributed by Clark Kimberling and Peter Moses, undated. As in the preamble just before X₅₀₁₉₉, in the plane of a triangle ABC, let

  P = point on Nagel line;

  D = point not on Nagel line or Euler line;

  U = point on Nagel line, other than U and G;

  L = line through U parallel to PD;

  U' = L^(Euler line).

  For centers X₅₀₂₃₇-X₅₀₂₄₄, we take P = X₁ and D = X₆. The appearance of (h,k), n in the following list means that if if U = h*a + k*(b+c) : : , then U' = X_n.

  (1,-2), 44217

  (1,-1), 377

  (1,0), 405

  (1,1) ,2

  (1,2), 50207

  (2,-1), 30

  (2,1), 50205

  [and others].

• X₅₀₂₄₇–X₅₀₂₅₅ — Points in a [[(b^2 - c^2, c^2 - a^2, a^2 - b^2 ], b^2 + c^2, c^2 + a^2, a^2 + b^2]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: (b^2 - c^2) α + (c^2 - a^2) β + (a^2 - b^2) γ = 0.

  L₂ is the line (b^2 + c^2) α + (c^2 + a^2) β + (a^2 + b^2) γ = 0.

  The origin is given by (0, 0) = X₃₈₅ = a^4 - b^2 c^2 : b^4 - c^2 a^2 : c^4 - a^2 b^2.

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = 2(a^4 - b^2 c^2) + (-2a^2 + b^2 + c^2) x + (b^2 - c^2) y : : ,

  where, as functions of a, b, c, the coordinate x is symmetric of degree 2, and y is antisymmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a b+a c+b c), 0}, 17731

  {-((2 a b c)/(a+b+c)), 0}, 19623

  {1/2 (-a^2-b^2-c^2), 0}, 15480

  {0, 0}, 385

  {1/2 (a^2+b^2+c^2), 0}, 230

  {a^2+b^2+c^2, 0}, 325

  [and others].

• X₅₀₂₅₆–X₅₀₂₇₈ — Points in a [[(b^2 - c^2, c^2 - a^2, a^2 - b^2 ], [(b^4 - c^4, c^4 - a^4, a^4 - b^4 ]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: (b^2 - c^2) α + (c^2 - a^2) β + (a^2 - b^2) γ = 0.

  L₂ is the line (b^4 - c^4) α + (c^4 - a^4) β + (a^4 - b^4) γ = 0.

  The origin is given by (0, 0) = X₂ =\(1 : 1 : 1\).

  Barycentrics \(u : v : w\) for a triangle center U = (x, y) in this system are given by

  \(u : v : w\) = (a^2 - b^2)(a^2 - c^2)(b^2 - c^2) + (-2a^2 + b^2 + c^2) x + (2a^4 - b^4 - c^4) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 4, and y is antisymmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c) (a+b+c), -((2 (a-b) (a-c) (b-c) (a+b+c))/(a^2+b^2+c^2))}, 50183

  {-((2 (a-b) (a-c) (b-c) (a^2+b^2+c^2))/(a+b+c)), -((2 (a-b) (a-c) (b-c))/(a+b+c))}, 50184

  {-((2 (a-b) (a-c) (b-c) (a b+a c+b c))/(a+b+c)), (2 (a-b) (a-c) (b-c))/(a+b+c)}, 50234

  {-((a-b) (a-c) (b-c) (a+b+c)), -((2 (a-b) (a-c) (b-c) (a+b+c))/(a^2+b^2+c^2))}, 50166

  {-(((a-b) (a-c) (b-c) (a^2+b^2+c^2))/(a+b+c)), -((2 (a-b) (a-c) (b-c))/(a+b+c))}, 49735

  [and others].

• X₅₀₂₈₁–X₅₀₃₁₆ — Points in a [[b-c,c-a,a-b ], [b^3 - c^3, c^3 - a^3, a^3 - b^3 ]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: (b-c) α + (c-a) β + (a-b) γ = 0.

  L₂ is the line (b^3 - c^3) α + (c^3 - a^3) β + (a^3 - b^3) γ = 0.

  The origin is given by (0, 0) = X₂ =\(1 : 1 : 1\).

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  \(u : v : w\) = (a-b)(a-c)(b-c)(a+b+c) + (-2a + b + c) x + (2 a^3 - b^3 - c^3) y ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 3, and y is antisymmetric of degree 3.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a-b) (a-c) (b-c), 0}, 3241

  {-2 (a-b) (a-c) (b-c), (2 (a-b) (a-c) (b-c))/(a b+a c+b c)}, 50133

  {-((2 (a-b) (a-c) (b-c) (a b+a c+b c))/(a^2+b^2+c^2)), (2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2)}, 1992

  {-((a-b) (a-c) (b-c)), -((2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2))}, 17274

  [and others].g

• X₅₀₃₁₈–X₅₀₃₂₅ — Points on the Euler line

  contributed by Clark Kimberling and Peter Moses, undated. In the plane of a triangle ABC, let

  P = point on Nagel line;

  D = point not on Nagel line or Euler line;

  U = point on Nagel line, other than U and G;

  L = line through U parallel to PD;

  U' = L^(Euler line).

  For centers X₅₀₁₉₉-X₅₀₂₀₈, we take P = X₈ and D = X₆. The appearance of (i,j) in the following list means that if if U = X_i then U' = X_j: (10,50318), (239,50319), (3187,50320), (3241,50321), (3621,50322), (3679,50323), (3811,50324), (4362,50325) [and others].

• X₅₀₃₂₆–X₅₀₃₅₉ — Points in a [[b c, c a, a b], [a^2, b^2, c^2]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: b c α + c a β + a b γ = 0.

  L₂ is the line a^2 α + b^2 β + c^2 γ = 0.

  The origin is given by (0, 0) = X₁₄₉₁ = a(b-c)(b^2+bc+c^2) : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  \(u : v : w\) = a(b - c)(b^2 + b c + c^2) - a(b - c) x - (b^2 - c^2) y ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 3, and y is antisymmetric of degree 3.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 (a^2+b^2+c^2), -2 (a^2+b^2+c^2)}, 47925

  {-2 (a b+a c+b c), -2 (a b+a c+b c)}, 47928

  {-((2 a b c)/(a+b+c)), -((2 a b c)/(a+b+c))}, 4490

  {-2 (a b+a c+b c), -a b-a c-b c}, 47666

  [and others]

• X₅₀₃₆₀–X₅₀₃₉₀ — Radical traces of circumcircle and other circles

  contributed by César Eliud Lozada, June 4, 2022. The appearance of (Ω, n) in the following list means that the radical trace of the circumcircle of ABC and circle Ω is X_n:

  (Adams circle, 50360), (anticomplementary circle, 858), (Apollonius circle, 50361), (Bevan circle, 1155), (Brocard circle, 187), (Conway circle, 50362), (Dao-Moses-Telv circle, 50363), (Dou circle, 9720), (Dou circles radical circle, 50364), (1st Droz-Farny circle, 44452), (Ehrmann circle, 6), [and others].

  Circles in cursive characters can be consulted in the Alphabetical Index of Terms in ETC (Alphabetical_Index.html). All other circles can be viewed in Wolfram’s Triangle Circles.

• X₅₀₄₄₉–X₅₀₄₅₉ — Points in a [[b c, c a, a b], [a^3, b^3, c^3]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: b c α + c a β + a b γ = 0.

  L₂ is the line a^3 α + b^3 β + c^3 γ = 0.

  The origin is given by (0, 0) = X₈₀₆₁ = a(b^4 - c^4) : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  \(u : v : w\) = a(b^4 - c^4) - a(b - c) x - (b^3 - c^3) y ,

  where, as functions of a, b, c, the coordinate x is symmetric of degree 3, and y is symmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-(a+b+c)^3, -((a b c)/(a+b+c))}, 47947

  {-((a+b+c) (a b+a c+b c)), -((a b c)/(a+b+c))}, 47959

  {-((a+b) (a+c) (b+c)), 0}, 661

  {-(1/2) (a+b) (a+c) (b+c), ((a+b) (a+c) (b+c))/(2 (a+b+c))}, 4129

  {0, 0}, 8061

  {0, ((a+b) (a+c) (b+c))/(a+b+c)}, 1577

  [and others].

• X₅₀₄₈₁–X₅₀₅₂₆ — Points in a [[b c, c a, a b], [b^2 c^2, c^2 a^2, a^2 b^2]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. If L₁ and L₂ are lines that meet in a point P not at infinity, then a [L₁, L₂]-coordinate system is a bivariate coordinate system having L₁ as x-axis, L₂ as y-axis, and P as origin. In this section, L₁ and L₂ are the following lines:

  L₁ is the line: b c α + c a β + a b γ = 0.

  L₂ is the line b^2 c^2 α + c^2 a^2 β + a^2 b^2 γ = 0.

  The origin is given by (0, 0) = X₆₄₉ = a^2 (b - c) : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  \(u : v : w\) = -a^3 b c (b-c) - a(b-c) x + a^2(b^2-c^2) y ,

  where, as functions of a, b, c, the coordinate x is symmetric of degree 4, and y is symmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-2 a b c (a+b+c), -((2 a b c)/(a+b+c))}, 47912

  {-2 (a b+a c+b c) (a^2+b^2+c^2), -2 (a^2+b^2+c^2)}, 47923

  {-2 (a b+a c+b c)^2, -2 (a b+a c+b c)}, 47926

  {-2 a b c (a+b+c), -a b-a c-b c}, 20983

  {-2 a b c (a+b+c), 0}, 4813

  [and others].

• X₅₀₅₃₃–X₅₀₅₃₆ — Points in an orthogonal [[b-c, c-a, a-b], [2 a^2 - 3a (b+c) - b^2 - c^2 + 6 b c, 2 b^2 - 3 b (c+a) - c^2 - a^2 + 6 c a , 2c^2 - 3 c (a+b) - a^2 - b^2 + 6 a b]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. The origin is given by (0, 0) = X₂ =\(1 : 1 : 1\) : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  \(u : v : w\) = -2(a^3 + b^3 + c^3 + 9 a b c - 2 a^2 b - 2 a b^2 - 2 b^2 c - 2 b c^2 - 2 c^2 a - 2 c a^2) + (-2 a + b + c) x + 3(b - c)(3 a - b - c) y : : ,

  where, as functions of a, b, c, the coordinate x is symmetric of degree 4, and y is symmetric of degree 4.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-a^2-b^2-c^2, 0}, 5121

  {0, 0}, 2}

  {2 (a^2+b^2+c^2), 0}, 5205}

  {-2*(a^2 + b^2 + c^2), 0}, X₅₀₅₃₃

  [and others].

• X₅₀₅₃₈–X₅₀₅₅₆ — Points in a [[a^4, b^4, c^4], [b^2 c^2, c^2 a^2, a^2 b^2]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. The origin is given by (0, 0) = X₅₀₅₄₉ = a^2 (b^6 - c^6) : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  a^2 (b^6 - c^6) + (b^4 - c^4) x + a^2 (b^2 - c^2) y : : ,

  where, as functions of a, b, c, the coordinate x is symmetric of degree 4, and y is symmetric of degree 4.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-a b c (a+b+c), (a b+a c+b c)^2}, 24290

  {1/2 (-a^4-b^4-c^4), 0}, 47126

  {1/2 (-a^4-b^4-c^4), 1/2 (a^2+b^2+c^2)^2}, 33294

  {0, 1/2 (a^4+b^4+c^4)}, 647

  [and others].

• X₅₀₅₇₅–X₅₀₆₃₈ — Points in a [[b-c, c-a, a-b], [b^2 c^2 (b^2 - c^2), c^2 a^2 (c^2 - a^2), a^2 b^2 (a^2 - b^2)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  U = a^2 (a-b)(a-c)(b-c)(b^2 + c^2 + a b + a c + b c) + (-2 a + b + c) x + a^2 (a^2 b^2 + a^2 c^2 - b^4 - c^4) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 6, and y is antisymmetric of degree 1.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-((a-b) (a+b) (a-c) (b-c) (a+c) (b+c)), 0}, 20018

  {-((a-b) (a-c) (b-c) (a^3+b^3+c^3)), ((a-b) (a-c) (b-c) (a+b+c))/(a b c)}, 3868

  {-a (a-b) b (a-c) (b-c) c, (2 (a-b) (a-c) (b-c))/(a^2+b^2+c^2)}, 3751

  {0, -(((a-b) (a-c) (b-c))/(2 (a b+a c+b c)))}, 4263

  [and others].

• X₅₀₆₄₅–X₅₀₆₈₆ — Intersection of radical axes involving apollonian circles

  :contributed by César Eliud Lozada, June 13, 2022. Let ABC be a triangle. Let A', A″ be the points at which the A-internal and the A-external angle bisectors cut BC. The circle with diameter A'A″ is called the A-apollonian circle of ABC, and the B- and the C- apollonian circles are built similarly.

  Let Ω be any other circle and let \(r_{a}\), \(r_{b}\), \(r_{c}\) be the radical axes of Ω and the A-, B- and C- apollonian circles of ABC, respectively. Then \(r_{a}\), \(r_{b}\), \(r_{c}\) concur in a point on the Brocard axis of ABC.

  If X, Y, Z are the powers of Ω with respect to A, B, C, respectively, then Q(Ω), the point of intersection of \(r_{a}\), \(r_{b}\), \(r_{c}\), is [barycentrics omitted here].

• X₅₀₇₇₁–X₅₀₇₇₆ — Points in a [[b^2 - c^2, c^2 - a^2, a^2 - b^2[, [a^2 - b^2 - c^2, b^2 - c^2 - a^2, c^2 - a^2 - b^2)]] coordinate system

  contributed by Clark Kimberling and Peter Moses, undated. The origin is given by (0, 0) = X₂₃₀ = 2 a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2 : : .

  Barycentrics \(u : v : w\) for a triangle center U = (x,y) in this system are given by

  U = 2 a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2 + (2 a^2 - b^2 - c^2) x + 2(b^2 - c^2) y : : ,

  where, as functions of a, b, c, the coordinate x is antisymmetric of degree 2, and y is antisymmetric of degree 2.

  The appearance of {x, y}, k in the following list means that (x, y) = X_k.

  {-((a^3+b^3+c^3)/(a+b+c)), -(((a-b) (a-c) (b-c))/(2 (a+b+c)))}, 24348

  {-a^2-b^2-c^2, 0}, 325

  {-(a+b+c)^2, 0}, 10026

  {-((a^3+b^3+c^3)/(a+b+c)), ((a-b) (a-c) (b-c))/(2 (a+b+c))}, 10

  {1/2 (-a^2-b^2-c^2), 0}, 44377

  {0, 0}, 230

  [and others].

• X₅₁₅₀₄–X₅₁₅₀₇ — Perspectors involving the García-Moses triangle

  This preamble is contributed by Clark Kimberling, based on notes from Emmanuel José García and Peter Moses. In the plane of a triangle ABC, let

  A'B'C' = cevian triangle of X₁; M_a = midpoint of AA', and define M_b and M_c cyclically; A″ = BM_c∩CM_b, and define B″ and C″ cyclicaly. The triangle A″B″C″ is here named the García-Moses triangle. Barycentrics for the vertices are given by

  A″ = a : a+c : a+b, B″ = b+c : b + b+a, C″ = c+b : c+a : c.

  The appearance of (T,i) in the following list means that T is perspective to A'B'C', and the perspector is X_i:

  (ABC, 10), (medial, 9), (excentral, 9), (extouch, 1), (1st circumperp, 4220), (half-altitude, 1), (1st Sharygin, 4220), (outer García, 10), (2nd extouch, 9), (tangential of excentral, 9), (García reflection, 9), (2nd Zaniah, 9), (Gemini 4, 9), (Gemini 5, 9), (Gemini 7, 2), (Gemini 8, 9), Gemini 15, 2), (Gemini 17, 2), (Gemini 25, 2), (Gemini 26, 10), (Gemini 119, 9), (extangents, 51504), (1st Conway, 51505), (anti-inner-García, 51506), (Aquilla, 51507)

  The appearance of (T,i) in the following list means that T is orthologic to A'B'C', and the orthology center is X_i:

  (Jenkins [centers of Jenkins circles], 10), (Ehrmann cross triangle, 30), (centers of the Apollonius circles, 511), (8th Brocard, 511), (1st Savin [c.f. X₄₄₃₀₁, 10)

  The appearance of (T,i) in the following list means that T is paralogic to A'B'C', and the parallelogy center is X_i:

  (Ehrmann cross triangle, 523), (centers of the Apollonius circles, 512), (8th Brocard, 512)

• X₅₁₆₁₅–X₅₁₆₁₈ — Incircle-inverses of points on the Euler line

  contributed by Clark Kimberling and Peter Moses, August 30, 2022. The incenter-inverse of the Euler line is the circle with center X₃₉₅₄₀ and pass-through points as indicated in the following list, in which the appearance of (i,j) means that X_i is on the Euler line, and X_j = incircle-inverse of X_i:

  (2,51615), (3,5590), (4,51616), (5,5533), (20, 51617), (30, 1), (1012, 51618), (1314,1315), (1315, 1314), (3109, 55509)

  See the preambles just before X₃₉₄₈₆, X₃₉₄₇₅, and X₅₁₆₁₉.

• X₅₁₆₁₉–X₅₁₆₃₁ — Circumcircle-inverses of points on the Nagel line

  contributed by Clark Kimberling and Peter Moses, August 30, 2022. The circumcircle-inverse of the Nagel line is the circle with center X₃₉₂₂₅ and pass-through points as indicated in the following list, in which the appearance of (i,j) means that X_i is on the Nagel line, and X_j = circumcircle-inverse of X_i:

  (1,36), (2,23), (8,17100), (10,1324), (42, 32759), (78, 51629), (386, 51619), (519, 3), (997, 51620), (1125, 51621), (3811, 51622), (5121,51623), (5205, 51630), (5529, 51624), (6788, 51625), (6789, 39225), (22837, 51626), (23869, 51627), (30144, 51628)

  See the preambles just before X₃₉₄₈₆, X₃₉₄₇₅, and X₅₁₆₁₅.

• X₅₁₆₃₂–X₅₁₆₃₉ — Circumcircle-inverses of points on the line X₁X_6

  contributed by Clark Kimberling and Peter Moses, August 31, 2022. The circumcircle-inverse of the line X₁X₆ is the circle with center X₃₉₂₂₇ and pass-through points as indicated in the following list, in which the appearance of (i,j) means that X_i is on X₁X₆, and X_j = circumcircle-inverse of X_i:

  (1,36), (6,187), (9,32625), (37,32758), (72,51632), (220,51633), (238,51634), (405,51635), (518,3) (956,51636), (958, 51637), (960, 51638), (5220,51639), (1001,5144), (1083,667), (3230,11650)

  See the preambles just before X₃₉₄₈₆, X₃₉₄₇₅, X₅₁₆₁₅, and X₅₁₆₁₉.

• X₅₁₆₄₀–X₅₁₆₆₄ — Points on the Helman line

  based on notes from Dan Reznik and Peter Moses, August 31, 2022. The radical axis of the incircle and circumcircle is here named the Helman line, after Mark Helman, who discovered special properties of this line under isgonal conjugation; see X₁₃₁₉. The Helman line passes through X₁₃₁₉ and is parallel to X₁X₃; the Helman line is also L(9), the trilinear polar of X₅₇, and it passes through X_i for these i:

  513, 663, 855, 1149, 1279, 1284, 1319, 1455, 1456, 1457, 1458, 1459, 1463, 1464, 1769, 2605, 3669, 3777, 4017, 4367, 4822, 6129, 6610, 6615, 14292, 14413, 25569, 39688, 42336, 43924, 48116, 48121, 48122, 48123, 48128, 48129, 48131, 48136, 48137, 48144, 48149, 48150, 48151, 48306, 48329, 48330, 48336, 48340, 48350, 48367, 48597, 48616, 50332, 50353, 50354, 50458, 50459, 50508, 50517, 50523, 50526, and 51640-to-51664.

Part 27

• X₅₂₃₀₂–X₅₂₃₄₃ — 4th tangent line ot two inconics

  contributed by César Eliud Lozada, November 5, 2022. Let ABC be a triangle, P₁ =\(x_{1} : y_{1} : z_{1}\), P₂ =\(x_{2} : y_{2} : z_{2}\) (barycentrics) two distinct points on the plane of ABC and not on the sidelines of ABC. Let \(q_{1}\), \(q_{2}\) be the inconics of ABC with perspectors P₁ and P₂, respectively.

  There exists a 4th-common tangent line τ to \(q_{1}\) and \(q_{2}\), other than the sidelines of ABC. It can be deduced that: The trilinear pole of τ has barycentrics:

  Q(τ) = \(x_{1}\)*\(x_{2}\)*(\(y_{1}\)*\(z_{2}\)-\(y_{2}\)*\(z_{1})\) : \(y_{1}\)*\(y_{2}\)*(\(x_{2}\)*\(z_{1}\)-\(x_{1}\)*\(z_{2})\) : \(z_{1}\)*\(z_{2}\)*(\(x_{1}\)*\(y_{2}\)-\(x_{2}\)*\(y_{1})\) The touchpoints T₁, T₂ of τ with \(q_{1}\), \(q_{2}\), respectively, are:

  T₁ = \(x_{1}\)*\(x_{2}\)^2*(\(y_{1}\)*\(z_{2}\)-\(y_{2}\)*\(z_{1})\)^2 : \(y_{1}\)*\(y_{2}\)^2*(\(x_{2}\)*\(z_{1}\)-\(x_{1}\)*\(z_{2})\)^2 : \(z_{1}\)*\(z_{2}\)^2*(\(x_{1}\)*\(y_{2}\)-\(x_{2}\)*\(y_{1})\)^2

  T₂ = \(x_{2}\)*\(x_{1}\)^2*(\(y_{2}\)*\(z_{1}\)-\(y_{1}\)*\(z_{2})\)^2 : \(y_{2}\)*\(y_{1}\)^2*(\(x_{1}\)*\(z_{2}\)-\(x_{2}\)*\(z_{1})\)^2 : \(z_{2}\)*\(z_{1}\)^2*(\(x_{2}\)*\(y_{1}\)-\(x_{1}\)*\(y_{2})\)^2

  The preceding coordinates have the following equivalences leading to geometric constructions of τ, T₁ and T₂: Q(τ) is the perspector of the circumconic of ABC passing through P₁ and P₂.

  Consequently, if U' and U″ are two given points and \(q_{0}\) is the circumconic {{A, B, C,U',U″}}, then, for any pair of points P', P″ on \(q_{0}\), the inconics with perspectors P', P″ have the same 4th common tangent line and it is tripolar of the perspector of \(q_{0}\).

  T₁ is the perspector of the cevian-triangle-of-P₁ and the side-triangle-of-(cevian-triangle-of-P₁ and cevian-triangle-of-P₂), which are always`perspective. (Note: the side-triangle A_sB_sC_s of two triangles A'B'C' and A″B″C″ has A-vertex A_s = B'C' ∩ B″C″ and cyclically B_s and C_s).

  Symmetrically, T₂ is the perspector of the cevian-triangle-of-P₂ and the side-triangle-of-(cevian-triangle-of-P₁ and cevian-triangle-of-P₂).

  See centers X₄₁₂₁₁ to X₄₁₂₂₄, which are related to analogous perspectors. T₁ is also the P₁-ceva conjugate of-Q(τ) and T₂ is also the P₂-ceva conjugate of-Q(τ), which means that T₁ is the perspector(cevian-of-P₁, anticevian-of-Q(τ)) and T₂ is the perspector(cevian-of-P₂, anticevian-of-Q(τ)).

  For shortening, the inconic with perspector U will be denoted in this section as IwP(U).

  The appearance of (i, j, m, n, u) in the following (reduced) list indicates that the 4th common tangent line of IwP(X_i) and IwP(X_j) touches them at X(m) and X_n, respectively, and has trilinear pole X(u): (1, 2, 244, 1015, 513), (1, 3, 2638, 3270, 652), (1, 4, 2310, 3270, 650), (1, 5, 41211, 41218, 2600), (1, 6, 3248, 1015, 649), (1, 7, 2310, 3022, 650), (1, 8, 2310, 3271, 650), (1, 10, 2643, 3122, 661), (1, 11, 52302, 52303, 46384), (2, 3, 35071, 2972, 520), (2, 4, 115, 125, 523), (2, 5, 39019, 35442, 6368), (2, 6, 1084, 3124, 512), [and others].

• X₅₂₄₄₄–X₅₂₄₄₇ — A locus related to K025

  contributed by César Eliud Lozada, November 15, 2022. Let ABC be a triangle and P a point on its plane. Let A'B'C' be the cevian triangle of P, A″ the reflection of A' in A. Let A* be the orthogonal projection of A″ in BC and build B*, C* cyclically. Then the locus of P such that ABC, A*B*C are perspective is the cubic K025 and the locus of the perspectors Q(P) is also K025. The appearance of (i, j) in the following list means that Q( X_i ) = X_j: (4, 4), (30, 265), (265, 30), (316, 671), (671, 316), (1263, 19552), (1300, 5962), (5080, 11604), (5134, 39993), [and others].

• X₅₂₄₄₈–X₅₂₄₅₆ — A locus related to K616

  contributed by César Eliud Lozada, November 15, 2022. Let ABC be a triangle and P a point on its plane. Let A'B'C' be the cevian triangle of P, A″ the midpoint of AA'. Let A* be the orthogonal projection of A″ in BC and build B*, C* cyclically. Then the locus of P such that ABC, A*B*C are perspective is the cubic K616 and the locus of the perspectors Q(P) is the cubic q’ = anticomplement-of-K1265, given by:

  q’: Σ[(SB*(S^2+(2*SB+SC)*SC)*y - SC*(S^2+(SB+2*SC)*SB)*z)*y*z] + (SB-SC)*(SC-SA)*(SA-SB) *x*y*z = 0

  q’ circumscribes ABC and passes through X_n for these n: {2, 4, 3434, 4846, 11185, 14593, 30513, 34029, 41370, 52448, 52449, 52450, 52451, 52452, 52453, 52454, 52455, 52456}.

  The appearance of (i, j) in the following list means that Q(X_i ) = X_j: (4, 4), (69, 2), (376, 4846), (1249, 52448), (3421, 30513), (5485, 11185), (6601, 3434), (9214, 52449), (34208, 14593), (36874, 52450), (36875, 52451), (36876, 52452), (36877, 52453), (36878, 52454), (51830, 52455), (51831, 41370), (51832, 52456)

• X₅₂₅₂₂–X₅₂₅₂₃ — Lachance-Moses-Briancon points

  based on notes from Michael Lachance and Peter Moses, November, 2022. In the plane of a triangle ABC, suppose that

  A' =\(a_{1} : a_{2} : a_{3}\)

  B' =\(b_{1} : b_{2} : b_{3}\)

  C' =\(c_{1} : c_{2} : c_{3}\)
  are points on the circumcircle.

  Let A_b = AB∩B'C' = (\(c^{2}\)\(b_{1}\) + \(a^{2}\)\(b_{3})\)\(c_{1}\) : \(a^{2}\)\(b_{3}\)\(c_{2}\) : 0, and define B_c and C_a cyclically.

  Let A_c = AC∩B'C' = (\(b^{2}\)\(c_{1}\) + \(a^{2}\)\(c_{2})\)\(b_{1} : 0 : a\)²\(b_{3}\)\(c_{2}\), and define B_a and C_b cyclically.

  Then the lines B_aC_a, C_bA_b, A_cB_c concur in the Brianchon point of the hexagon {{H(A_b, B_a, B_c, C_b, C_b, A_c, A_b)}}. This point, denoted by LMB(A, B, C,A',B',C'), is given by the following barycentrics:

  \(a_{2}\)\(a_{3}\)\(b_{1}\)\(c_{1}(b^{2}b_{3}c_{1} + c^{2}b_{1}c_{2} + a^{2}b_{3}c_{2})\) : :

  The following conic is inscribed in ABC:

  \(a^{4}\) \(b_{3}\)^2 \(c_{2}\)^2 (\(c^{2}\) \(a_{2}\) + \(b^{2}\) \(a_{3})\)² \(x^{2}\) + (cyclic) - 2 \(b^{2}\) \(c^{2}\) \(a_{2}\) \(a_{3}\) \(b_{1}\) \(c_{1}\) (\(a^{2}\) \(b_{3}\) + \(c^{2}\) \(b_{1})\)(\(a^{2}\) \(c_{2}\) + \(b^{2}\) \(c_{1})\) y z - (cyclic) = 0

  In particular, if A'B'C' is the circumtangential triangle, then LMB(A, B, C,A',B',C') = X₅₂₅₂₂, and the inscribed conic is the Kiepert parabola. If A'B'C' is the circumnormal triangle, then LMB(A, B, C,A',B',C') = X₅₂₅₂₃.

• X₅₂₅₂₆–X₅₂₅₄₈ — Centers of inscribed conics

  contributed by Peter Moses, December 9, 2022. In the preamble just before X₅₂₅₂₂, if A'B'C' is the cevian triangle of a point P =\(p : q : r\), then the center of the inconic is given by

  p (b^2c^2 p (q+r) + a (c^2q^2 + b^2r^2) : :

  The inconic is here given the name LMB-cevian-inconic of P. See also the preamble just before X₅₂₅₄₉.

• X₅₂₅₄₉–X₅₂₅₆₁ — Perspectors of inscribed conics

  contributed by Peter Moses, December 9, 2022. Referring to the preamble just before X₅₂₅₂₆, the perspector of the LMB-cevian-inconic is given by

  b^2*c^2*p*(b^2*p + a^2*q)*(c^2*p + a^2*r) : :

  See also the preamble just before X₅₂₅₄₉.

• X₅₂₇₂₆–X₅₂₇₄₀ — Clawson circles

  contributed by César Eliud Lozada, January 4, 2023. Let 𝓁₁, 𝓁₂, 𝓁₃, 𝓁₄ be four lines, no three of them being concurrent. Let T₁ be the triangle bounded by lines 𝓁₂, 𝓁₃, 𝓁₄, and denote T₂, T₃, T₄ similarly. Then the circumcenters of these four triangles lie on a circle Ω, here named the Clawson circle of the given lines. Moreover, the circumcircles of the four triangles have a common point 𝒬. (J. W. Clawson, #2898, The American Mathematical Monthly, Vol. 29, No. 5 (May, 1922), pp. 230-231.)

  The center of the circle Ω is the point QL-P5 in the Encyclopedia of Quadri-Figures.

  In this section, three of the lines are the sidelines of a reference triangle ABC and the fourth line is given. Assume this last line is the polar trilinear of P =\(x : y : z\), then the center of Ω is given by:

  Ω_o(P) = a^2*(x*(y^2-z^2)*(-a^2+b^2+c^2)+y*z*(-2*(b^2-c^2)*x+(a^2-c^2)*y-(a^2-b^2)*z)-(-a^2+b^2+2*c^2)*x^2*y+(-a^2+2*b^2+c^2)*z*x^2) : :

  and the point 𝒬, obviously on the circumcircle of ABC, is:

  𝒬(P) = a^2*(x-y)*(x-z) : :, which corresponds to the isogonal conjugate of - the infinite point of - the tripolar of - the isotomic conjugate of P.

  The appearance of (i, j) in the following list means that Ω_o(X_i) = X_j: (1, 52726), (3, 52737), (4, 44810), (5, 52738), (6, 14270), (7, 52730), (8, 52739), (9, 52740), (10, 44827), (25, 25644), (69, 8552), (76, 44826), (94, 14809), (107, 52734), (110, 52727), (111, 44821), (190, 52732), [and others].

• X₅₂₇₄₆–X₅₂₇₆₈ — Tripolar Centroidal Conjugates

  based on a definition and notes by Bernard Gibert; see Tripolar Centroidal Cubics. Suppose that P =\(p : q : r\) is a point in the plane of a triangle ABC. The tripolar centroidal conjugate of P is given by

  TCC(P) = (-2 p + q + r) (p q - 2 p r + q r) (-2 p q + p r + q r) : :

  and TCC(P) = P if and only if P lies on the cubic K015.

  Let U = the infinite point of the line GP, and let P' = tripolar centroid of P. Then TCCP is the trilinear pole of the line UP'.

• X₅₃₈₀₅ — Miyamoto-Moses Points

  Part 1 is based on notes and figures by Keita Miyamoto. Part 2 is based on notes from Keita Miyamoto and Peter Moses.

  Part 1.

  Proposition 1.1 (inner case).

  In a scalene acute triangle ABC, let M_aM_bM_c be its medial triangle. Let Ωa be the circle centered at M_a and passing through B and C, and define Ωb and Ωc cyclically. Inside ABC, let γa be the circle externally tangent to lines CA, AB,Ωa, and define γb and γc cyclically. Inside ABC, let γ be the circle internally tangent to Ωa, Ωb, Ωc. Then there exists a circle Γ that is tangent to the four circles γ, γa, γb, γc.

  Let Z₁ denote the touchpoint of Γ and γ, and let Z₂ denote the center of Γ. The points Z₁ and Z₂ are the triangle centers X₅₂₈₀₇ and X₅₂₈₀₅, respectively. Here, the circle Γ is named the 1st Miyamoto-Moses-Apollonius circle, and the circle internally tangent to γa, γb and γc is named the 2nd Miyamoto-Moses-Apollonius circle. Let T_a be the touchpoint of Γ and γa, and define T_b and T_c cyclically. Here, the triangle T_aT_bT_c is named the 1st-Miyamoto-Moses-Apollonius triangle. See Figure 1.

  Proposition 1.1 also holds when ABC is an obtuse triangle. In this case, suppose that angle A is obtuse. Let H_b be the point of intersection, other than A, of Ωc and line CA. Let H_c be the point of intersection, other than A, of Ωb and line AB. Let γa be the circle internally tangent to Ωa and segments AH_b and AH_c. This circle lies outside ABC, as in Figure 2.

  Proposition 1.2 (outer case).

  In a scalene acute triangle ABC, let M_aM_bM_c be its medial triangle. Let Ωa be the circle centered at M_a and passing through B and C, and define Ωb and Ωc cyclically. Let γa be the larger of the two circles internally tangent to lines CA, AB, Ωa, and define γb and γc cyclically. Outside ABC, let γ be the circle internally tangent to Ωa, Ωb, Ωc. Then there exists a circle Γ that is tangent to the four circles γ, γa, γb, γc.

  Let Z₃ denote the touchpoint of Γ and γ, and let Z₄ denote the center of Γ, as in Figure 3. The points Z₃ and Z₄ are the triangle centers X₅₂₈₁₀ and X₅₂₈₀₈, respectively. Here, the circle Γ is named the 3rd Miyamoto-Moses-Apollonius circle, and the circle internally tangent to γa, γb, γc other than Γ, is named the 4th Miyamoto-Moses-Apollonius circle. Let T_a be the touchpoint of Γ and γa, and define T_b and T_c cyclically. Here, the triangle T_aT_bT_c is named the 2nd Miyamoto-Moses-Apollonius triangle.

  Proposition 1.2 also holds when ABC is an obtuse triangle. In this case, suppose that angle A is obtuse. Let γa; be the smaller of the two circles, etc., as in Figure 4.

  It appears that Propositions 1.1 and 1.2 hold for any three circles Ωa, Ωb, Ωc, each of which passes through two of the three points A, B, C. Next, Proposition 1* treats an example of this sort.

  Proposition 1*.

  In a scalene acute triangle ABC, let Tan_aTan_bTan_c be the tangential triangle. Let Ωa be the circle centered at Tan_a and pssing through B and C, and define Ωb and Ωc cyclically. Let γa be a circle tangent (internally or externally) to lines CA, AB, Ωa, and define γb and γc cyclically. Let γ be the circle tangent to Ωa, Ωb, Ωc. Then there exists a circle Γ that is tangent to the four circles γ, γa, γb, γc.

  For Proposition 1*, there are 4 cases, as indicated by these figures: Figure 5, Figure 6, Figure 7, Figure 8. The 4 cases depend on internal and external tangencies.

  Proposition 2.

  Let γ be the incircle of ABC. Let γa be the circle through B and C and internally tangent to γ. Define γb and γc cyclically. Outside ABC, let Γa be the circle externally tangent to CA, AB, γa, and let A' be the center of Γa. Define Γb and Γc cyclically, and define B' and C' cyclically. Let L_a be the external common tangent, other than BC, of Γb and Γc, so that L_a is the reflection of BC in B'C'. Define L_b and L_c cyclically. Let A″=L_b∩L_c, and define B″ and C″ cyclically. The triangle A″B″C″ is perspective to ABC. The perspector is denoted by Z₅ in Figure 9. The point Z₅ is the triangle center X₅₂₈₁₇. Here, the triangle A'B'C' is named the Miyamoto-Moses triangle.

  Part 2.

  Barycentrics for points defined in Part 1 are shown below and in a sequel to Figure 1 in GeoGebra: Figure 10. This file has a slider that can be used to observe the effect of changing S to - S in the barycentrics. In particular, the transformation S -> - S takes X₁₄₁₂₁ to X₇₀₉₀.

  The point Z₅ shown in Figure 9, appears as P in Figure 11, and P = X₅₂₈₁₇.

  The circles γa, γb;, γc have a Chinese name that translates roughly as pseudo-circumcircles. For example, let U denote the circular hull of the pseudo-circumcircles that pass through 2 vertices and are tangent to the incircle. Their circular hull has center O = X₃ and radius R + r/2. In Figure 12, the points labeled tA and tA’ are given by barycentrics as follows:

  tA = 4*a^2*(a + b - c)*(a - b + c) : -((a - b - c)*(a + b - c)^3) : -((a - b - c)*(a - b + c)^3)

  tA’= 2*a^2 : a^2 - 2*a*b - b^2 - 2*a*c + c^2 : a^2 - 2*a*b + b^2 - 2*a*c - c^2

  If P =\(p : q : r\) is a point on the circumcircle, then the point

  P' = (a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4)*p + a^2*(a^2 - b^2 - c^2)*(q + r) : :

  lies on the the circular hull. The point P' is here named the Moses-Apollonius transform of P. Examples are X₅₂₈₂₀ to X₅₂₈₃₄.

• X₅₃₀₀₆–X₅₃₀₀₇ — Miyamoto-Moses Points

  This preamble continues the preamble just before X₅₂₈₀₅. It is based on proposition 3 by Keita Miyamoto, with barycentrics found by Peter Moses.

  Proposition 3.

  In a scalene acute triangle ABC, let T = T_aT_bT_c be the intouch triangle of ABC. Let γ be the incircle of T, and let Γa be the A-excircle of AT_bT_c. Define Γb and Γc cyclically. Let γa be the circle tangent to T_bT_c and internally tangent to the incircle of ABC at T_a. Define γb and γc cyclically. Then there exists a circle ω that is tangent to all seven of the circles γ, Γa, Γb, Γc, γa, γb, γc. Here the circle ω is named the Miyamoto-Moses circle.

  The center of Γa, which is the A-vertex of the 2nd midarc triangle (see X₁₀₄₉₁, given by

  a*b - b^2 + a*c + 2*b*c - c^2 - 2*x : b*(-a + b + c) : -((a - b - c)*c : : , where

  x = sqrt[b*c*(a + b - c)*(a - b + c)] = 2*b*c sin(A/2), and y and z are defined cyclically.

  The radius of Γa, is

  ((2*b*c + x)*S)/(2*b*c*(a + b + c)) = r*(1 + Sin[A/2])

  The touch-point of Γa and ω, denoted by a2 in Proposition 3 (GeoGebra)., is given by [expression omitted here].

  The center of the circle γa is given by

  2*a*x : 2*b*(a + b - c)*c + x*(-a + b + c) : 2*b*c*(a - b + c) + x*(-a + b + c)

  Let Z₆ be the touch-point of the circles ω and γ. Then Z₆ = X₅₃₀₀₆, and Z₆-of-T = X₁₃₅₇.

  Let Z₇ by the center of circle ω. Then Z₇ = X₅₃₀₀₇, and Z₇-of-T = X₅₃₀₀₂. The triangle a2b2c2 is perspective to ABC at X₁₀₄₈₉ and to the 2nd mid-arc triangle at X₅₃₀₀₇. The triangle a3b3c3 is perspective to the intouch triangle at X₁₀₄₈₉.

• X₅₃₂₄₆–X₅₃₅₈₀ — Intersections of lines tangent to conics

  contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), April 25-May 1, 2023. Suppose that U and X are distinct points on a conic sΓ. Let (U) be the line tangent to Γ at U, and Let (X) be the line tangent to Γ at X. Define \(F(U,X) = (U)\)∩(X). Listed here are conics for which F(U,X) appears for selected pairs of points U and X:

  circumcircle: X₅₃₂₄₆-X₅₃₃₃₀, X₅₃₃₈₄, X₅₂₃₈₅

  Steiner circumellipse: X₅₃₃₃₁-X₅₃₃₈₃

  Bevan circle: X₅₃₃₈₉-X₅₃₄₁₂

  Stevanovich circel: X₅₃₃₈₉, X₄₃₄₁₃

  other circumconics: X₅₃₃₈₆-X₅₃₃₈₉₈

  Kiepert circumhyperbola: X₅₄₃₁₄-X₅₄₅₂₀

  incircle: X₅₄₅₂₁-X₅₄₅₆₃

  nine-point circle: X₅₃₅₆₄-X₅₃₅₇₇

  Yff parabala: X₅₃₅₈₈-X₅₃₅₆₀₂

  dual of Yff parabola; i.e., {A, B, C,X₂, X₇}: X₅₃₅₈₈-X₅₃₅₆₀₂

  Suppose P =\(p: q : r\) is a point and Γ is the circumconic with perspector P. If U =\(u : v : w\) and X =\(x : y : z\), then the point \(f : g : h\) : F(U,X) is given by the following barycentrics:

  \(f : g : h\) : p(wy+vz)( (2qu+pv+qv-rv)x + (2pv+pu+qu-ru)y )( (2ru+pw+rw-qw)x + (2pw+pu+ru-qu)z ) : :

  In this case, \(F(U,X) = crossdifference of every pair of points on the line bcf\)α + gcaβ + habγ = 0, so that F(U,X) is the trilinear pole of the line ghα + hfβ + fgγ = 0 .

• X₅₃₆₀₃–X₅₃₆₁₃ — Reflection points on the circumcircle

  Clark Kimberling and Peter Moses May, 2023. The appearance of {i,j,k} in the following list means that X_k = reflection of X_i in the line X₃X_j:

  {74,98,43654}, {74,99,43654}, {74,100,43655}, {74,104,43655}, {74,110,74}, {74,1113,477}, {74,1114,477}, {74,1379,842}, {74,1380,842}, {74,1381,2687}, {74,1382,2687},

  {98,74,9161}, {98,99,98}, {98,110,9161}, {98,1113,842}, {98,1114,842}, {98,1379,2698}, {98,1380,2698}, {98,1381,2699}, {98,1382,2699}, {98,36735,953}, {98,36736,953}, [and many others],

• X₅₃₆₂₁–X₅₃₇₀₈ — and X₅₃₆₈₂-X₅₃₇₀₈, Touchpoints on the circumcircle

  contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), May 6, 2023. A point P, as a function of (a,b,c), is here defined to be an inner point if P lies inside the circumcircle, Γ, for every triangle ABC (that is, for every (a,b,c) satisfying b+c>a and c+a>b and a+b>c); and an outer point otherwise. If X is an outer triangle center (so that its Γ-invers is an inner triangle center), then X lies on two lines that are tangent to Γ, so that there are two touchpoints, which in some cases are triangle centers and in some cases are a bicentric pair. The appearance of X_i → (X_j,X_k in the following list means that X_i is an outer triangle center and X_j and X_k are the touchpoints

  X₁₉₀ → X(100, 932)

  X₃₅₁ → X(110, 111)

  X₆₅₉ → X(100, 105)

  [and othersj].

  In the analogous context for the Steiner circumellipse, instead of the circumcircle, see the preamble just before X₅₃₆₃₉.

• X₅₂₆₃₉–X₅₂₆₅₉ — Touchpoints on the Steiner circumellipse

  contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), May 7, 2023. In the manner of the preamble just before X₅₃₆₂₁, a point P, as a function of (a,b,c), is defined in this section to be an inner point if P lies inside the Steiner circumellipse, SCE, for every triangle ABC (that is, for every (a,b,c) satisfying b+c>a and c+a>b and a+b>c); and an outer point otherwise. If X is an outer triangle center (so that its SCE-inverse is an inner triangle center), then X lies on two lines that are tangent to SCE, so that there are two touchpoints, which in some cases are triangle centers and in some cases are a bicentric pair. The appearance of X_i → (X_j,X_k in the following list means that X_i is an outer triangle center and X_j and X_k are the touchpoints

  X₁₀₀ → X(190, 664)

  X₁₀₇ → X(648, 53639)

  X₁₁₀ → X(99, 648)

  [and others].

  In the analogous context for the circumcircle, instead of the Steiner circumellipse, see the preamble just before X₅₃₆₂₁.

• X₅₃₆₆₀–X₅₃₆₇₁ — Strictly inner points

  contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), May 8, 2023. In the preamble just before X₅₃₆₂₁, a point P is defined to be an inner point if P lies inside the circumcircle for all triangles ABC. Here, a point P is defined to be a strictly inner point point if P lies inside ABC for every nondegenerate triangle ABC. That is, if b+c>a and c+a>b and a+b>c, then P has barycentrics (and trilinears) that are all positive for all such (a,b,c). Many well-known triangle centers have this property (e.g., X₁, X₂, X₆, X₉, X₁₀, X₃₇.) A point P is a strictly outer point point if P lies outside ABC for every nondegenerate triangle ABC.

• X₅₃₇₀₉–X₅₃₇₆₃ — Midpoints of points on the circumcircle

  contributed by Clark Kimberling and Peter Moses, May 10, 2023. The points all lie inside the circumcircle, Γ, so that, they are “inner points” as defined in the preamble just before X₅₃₆₂₁. Conjecture: none of these midpoints is “strictly inner”; i.e., inside triangle ABC for all A, B, C.

  Suppose that P is a fixed point on Γ and that X is a variable point on Γ. The locus of the midpoint of P and X is a circle, Γ(P,X). Let O(P,X) be the center of Γ(P,X). For example, O(X₇₄,X) passes through X_i for these i: 12041, 12042, 33813, 33814, 38599, 38600, 38601, 38602, 38607, 1511, 14650, 38608, 38609. If P is not fixed but goes around the circle, then the locus of the centers of the circles has center O and radius (1/2)*circumradius. This circle, with perspector X₃₄₃₁, passes through X_i for these i: 1511, 12041,12042, 14650, 33813, 33814, 35231, 35232, 38599, 38600, 38601, 38602, 38603, 38604, 38605, 38606, 38607, 38608, 38609, 38610, 38611, 38612, 38613, 38614, 38615, 38616, 38617, 38618, 38619, 38620, 38621, 38622, 38623, 38624, 38625, 49119

Part 28

• X₅₄₀₉₇–X₅₄₁₁₃ — Perspectors associated with Steiner-circumcevian triangles

  contributed by Clark Kimberling and Peter Moses June 29, 2023. Let SCE be Steiner circumellipse of a triangle ABC, and Let P =\(p : q : r\) be a point not on a sideline of ABC. Let A' be the point, other than A, where the line AP meets line BC. Define B' and C' cyclically. The triangle A'B'C' is here named the Steiner-circumcevian triangle of P, denoted by SCC(P). The vertices of SCC(P) are given by

  A' = -q r : q (q + r) : r (q + r)

  B' = p (r + p) : - r p : r (r + p)

  C' = p (p + q) : q (p + q) : - p q

  The triangle SCC(P) is perspective to the anticomplementary. Let SC(P) denote the perspector. Then

  SC(P) = \(P^{2}\)-Ceva conjugate of X₂

  SC(P) = anticomplement of isotomic conjugate of \(P^{2}\).

  The appearance of (i,j) in the following list means that SC(X_i) = X_j.

  (2996, 54097), (330, 54098), (7035, 54099), (276, 54100), (291, 54101), (514, 54102). (8781, 54103), (523, 54104), (40410, 54105), (262, 54106), (333, 54107), (18020, 54108), (314, 54109), (4998, 54110), (253, 54111), (310, 54112), (312, 54113)

  The triangle SCC(X₂) is the triangle Gemini 107, and the triangle SCC(4) is the 9th Brocard triangle.

  Let T_a be the line tangent to SCC at A', and define T_b and T_c cyclically. Let A* = T_b ^ T_c and define B* and C* cyclically. Then A*B*C* is perspective to ABC, and the perspector is given by the point T(P) = 1 / (-q*r + r*p + p*q) : : . The transformation T maps curves to curves, as in these examples:

  T(Kiepert hyperbola) = Kiepert hyperbola

  T(K184) = K007

  T(K1023) = K1000

  T(K868) = K1002

  T(K1014) = K1037)

  T(K342a) = K1053a)

  T(K342b) = K1053n)

• X₅₄₄₄₉–X₅₅₀₃₇ — and X₅₅₀₁₉-X₅₅₀₃₇, Cyclocevian conjugates

  contributed by Clark Kimberling and Peter Moses, July 11, 2023. As noted in the Glossary, suppose that P =\(p : q : r\) (trilinear coordinates, not barycentric) is a point not on a sideline of ABC, and let A'B'C' be the cevian triangle of P. The circumcircle of A'B'C' meets line BC in two points: A' and A″; pairs B', B″, and C',C″ are obtained cyclically. The lines AA″, BB″, CC″ concur in the cyclocevian conjugate of P. Let

  \(g(a,b,c) = a/[p(qb + r_{c})]\) and \(f(a,b,c) = bc/[g(b\),c,a) + g(c,a,b) - g(a,b,c)].

  The cyclocevian conjugate of P is given by

  \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (trilinears).

  The cyclocevian conjugate of a point is the

  isotomic conjugate

  of the anticomplement

  of the isogonal conjugate

  of the complement

  of the isotomic conjugate

  of the point

  (Darij Grinberg, January 24, 2003)

  Now switching to bartycentric coordinates, suppose that p x + q y + r z = 0 is a line. It’s image under cyclocevian conjugationj is the octic curve given by [expression omitted here]

  For example, the cyclocevian image of the Euler line passes through A, B, C, and the anticevian triangle of ABC, and through X_i for these i: 2, 4, 1032, 13580, 13581, 54449.

  The appearance of (i,j) in the following list means that the cyclocevian conjugate of X_i is X_j: (1,1029, (2,4), (5,54449), (6,1031), (7,7), (8,189), (13,13483), (14,13484), (20,1032), (63,54450), (66,2998), (67,46275), (68,34287), (69,253), (75,8044), (76,41513), [and others].

  If “circumcircle” is replaced by “Steiner circumellipse” in the definition of cyclocevian conjugate, the result is here named the Steiner-cevian conjugate of X. The appearance of (i,j) in the following list means that the Steiner-cevian conjugate of of X_i is X_j:

  (1,13610), (2,2), (4,43710), (6,14370), (7,43750), (8,7155), (69,43714), (75,18298), (148,31998), (192,3212), (194,3186), (513,9267), (514,42555), (523,9293), [and others]. If “circumcircle” is replaced by “Kiepert circumhyperbola” in the definition of cyclocevian conjugate, the result is here named the Kiepert-cevian conjugate of X. The appearance of (i,j) in the following list means that the Kiepert-cevian conjugate of of X_i is X_j: (1,13486), (2,99), (3,110), (4,35360), (6,13578), (13,36839), (14,36840), (30,476), (399,47053), (616,35314), (617,35315), (5667,4240}

• X₅₄₄₆₀–X₅₄₄₆₇ — H-conics

  contributed by César Eliud Lozada, July 15, 2023. Let ABC be a right triangle at A. There exists a lot of finite centers in ETC lying on the hypotenuse BC (produced included), in particular, those having first coordinates with a multiplying factor cos(A) or (-a^2+b^2+c^2).

  Let denote by ℋ the set of k such that X_k-of-ABC lies on the hypotenuse BC. As an example, the subset o ℋ for k≤1000 is {3, 48, 49, 63, 68, 69, 71, 72, 73, 77, 78, 97, 122, 123, 125, 127, 130, 131, 155, 184, 185, [and others].

  Application.

  Let ABC be an acute triangle . Build the rectangle \(BCC_{a}\)B_a such that A lies on C_aB_a. Two right triangles B_aBA and C_aCA are obtained. Now, for a given k ∈ ℋ, let B'_a = X_k-of-B_aBA and C'_a = X_k-of-C_aCA, these centers lying on their hipotenuses AB and AC, respectively. Define C'_b, A'_b and A'_c, B'_c cyclically. It is not hard to prove that, for any k∈ℋ, these six points lie on an conic 𝒞( X_k ), here named the H-conic of X_k (H stands for hypotenuses).

  Depending on the chosen k, 𝒞( X_k ) can degenerate to two lines or to the line at infinity (as with X₃, X₆₈ and others). Also, every pair of constructed points on a side of ABC can coincide and the H-conic approaches to a circle, as occurs with X₆₉, for which the Taylor circle is obtained.

• X₅₄₄₇₄–X₅₅₀₀₉ — Orthology centers related to bicevian conics

  contributed by Ivan Pavlov, July 19, 2023. Let (c) be the bicevian conic of P={u,v,w} and Q={p,q,r}. Lines AP, BP, CP intersect (c) at six points, three of which form the cevian triangle of P. Denote the other three with A₁, B₁, and C₁. Similarly, using Q, define A₂, B₂, and C₂. The lines A₁A₂, B₁B₂, and C₁C₂ form a triangle T_aT_bT_c, which is always perspective to ABC.

  In the cases when a certain fourth degree relation holds, ABC and T_aT_bT_c are also orthologic. In the particular case when Q=X₂ and P lies on the Kiepert hyperbola, the orthology center of ABC and T_aT_bT_c also lies on the Kiepert hyperbola.

  In the particular case when Q=X₄ and P lies on the circumconic with perspector X₄, the configuration is degenerate becasue T_a=T_b=T_c=H. The orthology center (which exists only in the limit) is the isotomic conjugate of (SB u v+SC u w-a^2 v w : :) and lies on the Steiner circumellipse. When the orthology center of ABC and T_aT_bT_c exists it lies on the Euler line.

  For more information on how each center arises see the documents attached to Euclid 5932 .

  Contributed by Peter Moses, July 21, 2023: The appearance of i in the following list means that X_i is a major center and lies on the Kiepert hyperbola: 54479, 54480, 54534, [and others].

• X₅₅₀₄₂–X₅₅₀₇₃ — Centers of circumconics

  contributed by Clark Kimberling and Peter Moses July 25, 2023. In the plane of a triangle ABC, let P =\(p : q : r\) and U =\(u : v : w\) be distinct points. The center of the circumconic {{A, B, C,P,U}} is given by

  p u (r v - q w)(p v w (r - q) - q w u (p + r) + r u v (p + q ) : :

  See X₃₄₅₈₅.

• X₅₅₁₇₉–X₅₅₂₈₅ — Tripoles of mixed polar lines

  contributed by Ivan Pavlov, August 3, 2023. Let P and Q be two points and C_P and C_Q their polar conics in the cubic K . The polar lines of P in C_Q and Q in C_P coincide. This common polar line is here introduced as the mixed polar line of P and Q in K.

  In general, for the cubic k1 x^2 y + k2 x y^2 + k3 x^2 z + k4 x z^2 + k5 y^2 z + k6 y z^2 + k7 x^3 + k8 y^3 + k9 z^3 + k10 x y z, the mixed polar line of {u,v,w} and {p,q,r} is given by these coefficients:

  {2*(3*k7*p+k1*q+k3*r)*u+(2*k1*p+2*k2*q+k10*r)*v+(2*k3*p+k10*q+2*k4*r)*w,

  (2*k1*p+2*k2*q+k10*r)*u+2*(k2*p+3*k8*q+k5*r)*v+(k10*p+2*k5*q+2*k6*r)*w

  (2*k3*p+k10*q+2*k4*r)*u+(k10*p+2*k5*q+2*k6*r)*v+2*(k4*p+k6*q+3*k9*r)*w}

  In this section, we consider some mixed polar lines wrt K001 Neuberg cubic and K002 Thomson cubic.
  Examples of {m, n, l} for which the mixed polar line of X(m) and X_n in K001 is the tripolar of the isotomic conjugate of X(l) follow:

  {1,30,32679}; {2,3,31072}; {2,20,12077}; {2,30,3268}; {3,6,23285}; {3,30,8552}; {4,30,44427}; {5,30,46603}; {6,30,526}

  Following are some examples of {m, n, l} for which the mixed polar line of X(m) and X_n in K002 is the tripolar of the isotomic conjugate of X(l):

  {1,2,661}; {1,6,4374; {1,8,48334}; {1,9,20906}; {1,10,48131}; {1,42,47672}; {1,43,693}; {1,44,21433}; {1,46,23685}; {1,57,21438}; {1,63,20909}; {1,200,48398}

• X₅₅₃₂₁–X₅₅₃₈₄ — Touchpoints conics

  contributed by César Eliud Lozada, August 5, 2023. Denote by q(P) the circumconic of ABC with perspector P with respect to ABC, and consider two circumconics q’ = q(P') and q” = q(P″). If P' and P″ lie both in the interior of ABC, these conics have four real intersections {A, B, C, D} (D being the tripole of the line P'P″) and four common tangents. Assume P' =\(U' : V' : W'\) and P″ = U″ : V″ : W″ (trilinears, for simpler expressions). The following results were found algebraically: [lists omitted here].

• X₅₅₄₈₃–X₅₅₄₉₅ — Centers on the cubic K005

  contributed by César Eliud Lozada, August 9, 2023. Most of these centers are the 3rd intersection of K005 and the line {P, Q}, where P and Q lie on K005.

• X₅₅₄₉₆–X₅₅₅₂₈ — Centers on the cubic K006

  contributed by César Eliud Lozada, August 9, 2023. Most of these centers are the 3rd intersection of K006 and the line {P, Q}, where P and Q lie on K006.

• X₅₅₈₃₀–X₅₅₈₃₇ — Centers on the cubic K007

  contributed by César Eliud Lozada, August 12, 2023.

• X₅₅₈₅₅ — Centers X(55838-X₅₅₈₅₅, Centers on the cubic K008

  contributed by César Eliud Lozada, August 12, 2023. For centers on the cubic K008 (Part 2), see Centers X₅₆₄₇₁-X₅₆₄₉₄.

• X₅₅₉₁₇–X₅₆₃₆₅ — Kimberling-Pavlov conjugates

  contributed by Ivan Pavlov, August 14, 2023. Let P₁={a1,a2,a3} and P₂={b1, b2, b3) be arbitrary points and let (cc) be the circumconic with perspector X={u,v,w}. Let A₁, B₁, C₁ and A₂, B₂, C₂ be the traces on (cc) of P₁ and P₂, respectively.

  The lines A₁A₂, B₁B₂, C₁C₂ form a triangle perspective to ABC. The perspector has the following barycentrics:

  u/(u^2/(a1*b1)-(v/a2+w/a3)*(v/b2+w/b3)) : v/(v^2/(a2*b2)-(u/a1+w/a3)*(u/b1+w/b3)) : w/(w^2/(a3*b3)-(u/a1+v/a2)*(u/b1+v/b2))

  This point is here introduced as the Kimberling-Pavlov X-conjugate of P1 and P2. It is obviously symmetric and involutory (i.e., it is a conjugation). In his article “Mappings Associated with Vertex Triangles” (:cite:`Forum Geometricorum <forum-geometricorum>`, 9 (2009) 27-39), Clark Kimberling discusses this mapping for the case X=X₆, and he denotes the mapping by M₁. He also proposes variations denoted by M₂, M₃, and M₄.

  Here, these mappings are generalized for any point X, and the equivalent of formula (6) on p.34 of the cited article gives the following barycentrics:

  (KP2(X) of P₁ and P₂) = u/(u^2/(a1*b1)-(v/a2-w/a3)*(v/b2-w/b3)) : v/(v^2/(a2*b2)-(u/a1-w/a3)*(u/b1-w/b3)) : w/(w^2/(a3*b3)-(u/a1-v/a2)*(u/b1-v/b2))

  (KP3(X) of P₁ and P₂) = u/(u^2/(a1*b1)+(v/a2+w/a3)*(v/b2+w/b3)) : v/(v^2/(a2*b2)+(u/a1+w/a3)*(u/b1+w/b3)) : w/(w^2/(a3*b3)+(u/a1+v/a2)*(u/b1+v/b2))

  (KP4(X) of P₁ and P₂) = u/(u^2/(a1*b1)+(v/a2-w/a3)*(v/b2-w/b3)) : v/(v^2/(a2*b2)+(u/a1-w/a3)*(u/b1-w/b3)) : w/(w^2/(a3*b3)+(u/a1-v/a2)*(u/b1-v/b2))

  Stated below are a few properties of these points:

  Theorem 1.

  Let I=X₁ and

  P'= cevapoint of I and the isogonal conjugate of P

  Q'= cevapoint of I and the isogonal conjugate of Q.

  Then the Kimberling-Pavlov I-conjugate of P and Q is the intersection, other than A, B, C, of the conics {{A, B, C,P,Q'}} and {{A, B, C,P',Q}}.

  Theorem 2.

  In the limiting case, where P=Q, the Kimberling-Pavlov I-conjugate of P and P is the cross-conjugate of I and the the isogonal conjugate of P(P).

  Generally, the Kimberling-Pavlov X-conjugate of P and P is the cross conjugate of the X^2-reciprocal conjugate of P and P, where “ ^ “ denotes barycentric square.
  Theorem 3.

  The Kimberling-Pavlov X₆-conjugate of P and Q is the P-vertex conjugate of Q.
  Theorem 4.

  Let 𝓒 be a circumconic through I. If P lies on 𝓒, then the Kimberling-Pavlov I-conjugate of I and P also lies on 𝓒.
  Theorem 5.

  The Kimberling-Pavlov G-conjugate of P and Q is the isotomic conjugate of the midpoint of the barycentric quotients P/G and Q/G.

Part 29

• X₅₆₄₇₁–X₅₆₄₉₄ — Centers on the cubic K008 (Part 2)

  contributed by César Eliud Lozada, August 17, 2023. For centers on the cubic K008 (Part 1), see Centers X₅₅₈₃₈-X₅₅₈₅₅.

• X₅₆₅₆₉–X₅₆₇₃₀ — Centers on selected cubics

  contributed by César Eliud Lozada, August 19, 2023.

• X₅₇₀₄₂–X₅₇₂₅₂ — Triaxial points

  contributed by Ivan Pavlov, August 26, 2023. Triaxial points are defined in the preamble just before X₁₄₂₇₂.

  “Let F₁, F₂, F₃ be three figures in perspective two and two in the same plane, show that if they have a common centre of perspective, their three perspectrix are concurrent.” (Quoted from Lachlan, R.: An Elementary Treatise on Modern Pure Geometry, McMillan & Co., 1893, pp. 123). Let T(P,Q) denote the triaxial point of ABC, the P-circumconcevian triangle of X={u,v,w}, and the Q-circumconcevian triangle of X. Ten T(P,Q) depends only on the line PQ and coincides with

  (1) the cevian quotient of the tripole of line PQ and X, and

  (2) the pole of line PQ wrt the circumconic with perspector X

  In barycentrics, if P =\(p1 : q1 : r1\) and p2 =\(p2 : q2 : r2\), then

  T(P,Q) = u ((q2 r1 - q1 r2) u + (p2 r1 - p1 r2) v + (p1 q2 - p2 q1) w) : v ((q1 r2 - q2 r1) u + (p1 r2 - p2 r1) v + (p1 q2 - p2 q1) w) : w ((q1 r2 - q2 r1) u + (p2 r1 - p1 r2) v + (p2 q1 - p1 q2) w).

• X₅₇₂₉₇–X₅₇₃₈₁ — Homothetic centers involving nine-point centers

  contributed by Ivan Pavlov on August 30, 2023. Let P=\(u : v : w\) be a point upon the circumcircle of ABC and let P' be the antipoade of P. Let O_a, O_b, O_c be the nine-point centers of BPC, ACP, ABP resp.

  Triangle O_aO_bO_c is homothetic to ABC. The center of homothety is PX₅∩P'X₁₄₀, given by the following first barycentric:

  a^4*u+(b^2-c^2)^2*(2*u+v+w)-a^2*(b^2+c^2)*(3*u+v+w).

Part 30

• X₅₈₂₆₉–X₅₈₂₇₈ — Points on the Yff hyperbola

  contributed by Clark Kimberling and Peter Moses, undated.:A barycentric equation for the Yff hyperbola (MathWorld) is the following: [long equation omitted here]. The center of the Yff hyperbola is X₃₈₁, and the hyperbola passes through X_i for i = 2, 4, 14163, 14164, 14214, 14215, and 58269-58278. See PU(213) in Bicentric Pairs. The perspector of the Yff hyperbola is X₁₃₄₈₁.

• X₅₈₇₄₇–X₅₈₉₀₉ — Tripolar triangles

  contributed by César Eliud Lozada, September 24, 2023. Let T' = A'B'C', T″ = A″B″C″ be two perspective triangles, neither inscribed in the other. Denote their perspector Q and their axis of perspectivity r.

  Let A₁, B₁, C₁ be the tripoles of B'C', C'A', A'B' with respect to T″, in the same order. The triangle A₁B₁C₁ is named here the tripolar triangle of T’ with respect to T” and denoted as TPT(T', T″).

  Let A₂B₂C₂ be the tripolar triangle of T″ with respect to T', i.e, T₂=TPT(T″, T')

  Then: .LI{padding-bottom:12px;} a) T' and TPT(T″, T') are perspective. Call their perspector Q' and their perspectrix r’. The point Q' is referred here as the tripolar-perspector of T’ to T”. b) T″ and TPT(T', T″) are perspective. Call their perspector Q″ and their perspectrix r”. c) Q, Q', Q″ are collinear on a line L. d) r, r’, r” concur in a point, here named the tripolar-triaxial point of T’ and T”. e) Call P', P″ the tripoles of L wrt T', T″, respectively. Then Q, P', P″ are collinear.

  A list of related centers for triangles ABC and orthic can be seen here (X(58747).txt).

  Note: For definitions of triangles used in this section, check the Index of triangles referenced in ETC (IndexOfTrianglesReferencedInETC.html).

• X₅₉₁₃₇–X₅₉₂₇₃ — Polelogic and polarologic centers involving circumcevian triangles

  contributed by Ivan Pavlov on September 26, 2023. The polelogic and polarologic centers of two triangles T' and T″ are defined in the preamble just before X₄₂₂₈₇. It can be shown that T' = ABC and the circumcevian triangle (denoted T″ below) of any point P =\(u : v : w\) are polelogic.

  T'-to-T″-polarologic center = a^2 c^2 u v - a^4 v w + b^2 u (2 c^2 u + a^2 w) : :

  > T'-to-T″-polelogic center = u (b^2 u + a^2 v)(c^2 u + a^2 w)(-c^4 u v + c^2 (b^2 u + a^2 v) w + 2 a^2 b^2 w^2)(-2 a^2 c^2 v^2 + b^4 u w - b^2 v (c^2 u + a^2 w)) : :

  T″-to-T' polarologic center = X₆,

  T″-to-T' polelogic center = b^2 c^2 u (b^2 u + a^2 v)(c^2 u + a^2 w) : : .

  These centers can be generalized by reference to circumconcevian triangles instead of circumcevian triangles. See Euclid 5406 for details.

• X₅₉₂₉₄–X₅₉₃₅₉ — Harmonic means

  contributed by César Eliud Lozada, September 30, 2023. The harmonic mean x of two numbers p, q is that satisfying \(x^{-1}\) = (\(p^{-1}\) + \(q^{-1})\)/2. Similarly, in geometry, given three collinear points O, P, Q, their harmonic mean is the point X, on its same line, and such that OX is the harmonic mean of OP and OQ (signed distances all).

  If O, P, Q are given and X is their harmonic mean, then X is the point for which (O, X) and (P, Q) are in harmonic range. X is denoted here as the O-harmonic mean of (P, Q).

  A very simple geometrical construction of X = O-harmonic mean of (P, Q) follows, when O, P, Q are given: .LI li{padding-bottom:12px;} Take any point E outside the line OPQ and join OE and QE. Through P, draw a parallel line to QE, cutting OE at F. Let F* be the reflection of F in P. The line EF* cuts the line OPQ in X, the O-harmonic mean of (P, Q).

  Algebraically, if λ = OQ/OP, then OX = (2*λ/(1+λ))*OP.

• X₅₉₄₃₇–X₅₉₄₅₈ — Inconics through two given points

  contributed by César Eliud Lozada, October 5, 2023. In Eagles, Thomas Henry, Constructive Geometry of Plane Curves, McMillan and Co., London, 1885, available in www.archive.org, there are expained and proved methods for constructing conics touching three distinct, not concurrent given lines and passing through two given points (problem 83, pp.134 and problem 112, pp. 182).

  Let ABC be the triangle bounded by the three given tangents and P, Q the given points. Both P and Q must be in the same side with respect to every tangent. This means that P and Q must be both interior to ABC (problem 83) or both exterior to it (problem 112).

  The construction by Eagles follows (using century XXI cyclic notation): .LI li{padding-bottom:12px;} Let A', B', C' be the intersections of line PQ and BC, CA, AB, respectively. Let B₁, B₂ be the double points of the involution {A', B'} and {P, Q} (see preamble just before X₁₄₇₈₂). Eagles call these double points the foci of the involution and he proves that these points lie on the chord joining the touchpoints of the inconic with BC and AC. Analogously, let C₁, C₂ be the double points of the involution {A', C'} and {P, Q}. These points lie on the chord joining the touchpoints of the inconic with BC and AB. Take any point obtained in 2), for example B_i=B₁, and any point obtained in 3), for example C_j=C₁, and let H_a be the A'-harmonic mean of (B_i, C_j). (See preamble just before X₅₉₂₉₄) The line AH_a cuts BC at A*, which is the touchpoint of the desired inconic with the line BC. Then, from 2) and 3), the other touchpoints of the inconic are B* = A*B_i∩AC and C* = A*C_j∩AB. Now we have either five points on the conic or can get the perspector of the inconic; then the conic is unambigously determined. Repeat 4) and 5) with the other possible combinations of B_i and C_j and, finally, four conics are found, all touching the given lines and passing through the given points.

  Some remarks are needed here: The above construction was made by starting from A'. If we start from B' or C', the same set of inconics is obtained in each case. If P, Q are both interior to ABC then the four inconics are ellipses, but if they are both exterior to ABC then the resulting inconics can be of mixed types, i.e., some of them can be ellipses and some of them can be hyperbolas. Eagles does not mention this plurality in problem 112. When P, Q are triangle centers or when they are a bicentric pair, it can be conjectured that just one out of the four inconics is a central inconic, which means that its perspector and center are both triangle centers. Also, it can be conjectured that the two triads of perspectors and centers of the remaining non-central inconics are, respectively, vertices of two central triangles. It is necessary to examine which combination of B_i and C_j leads to the central inconic, because the required combination depends on the relative positions of P and Q.

  The appearance of (i, j, m, n) in the following list means that the central inellipse through X_i and X_j has perspector X(m) and center X_n: (1, 2, 59437, 59438), (1, 6, 59439, 59440), (1, 7, 59441, 59442), (1, 8, 59443, 59444), (1, 31, 765, 24036), (1, 32, 59445, 59446), (2, 3, 59447, 59448)^*, (2, 6, 1016, 4422), (2, 7, 59449, 59450), (2, 8, 55339, 59451), (2, 31, 59452, 59453), (2, 32, 4590, 620), (3, 4, 55346, 15252)^*, (7, 8, 4998, 3035), (7, 55, 59457, 59458), (7, 56, 4998, 3035), (8, 55, 4076, 3039), (8, 56, 4998, 3035), (31, 32, 59455, 59456), (55, 56, 59, 13006)

  ^(*): For acute ABC only. This fact guarantees that X₃ and X₄ are interior to ABC. Note that all the other centers X_i, X_j used in this list are always interior to any acute or obtuse triangle.

  The inellipse through the Brocard points (PU(1)) has perspector X₅₂₂₀₅ and center X₅₉₄₅₄. The inellipse through their isotomic conjugates, PU(11), has perspector X₄₀₀₉₈ and center X₂₆₅₈₂.

  A general expression for the perspectors of these inconics can be seen here (X(59437).pdf).

• X₅₉₄₅₉–X₅₉₄₈₃ — Lateral-inconics and related triangles

  contributed by César Eliud Lozada, October 8, 2023. Continuing from the preamble just before X₅₉₄₃₇, consider the three non-central inconics of ABC passing through two given centers U, X.

  Ordered properly, these inconics seem to be related each to one vertex of ABC. In this section, the A-, B-, C- inconics through U, X wil be referred as the lateral-inconics of (U,X). Let T_p be the triangle with vertices in the perspectors of the lateral inconics of (U, X) and T_c the triangle with vertices on their centers.

  For all centers considered in this section, it resulted that T_p is perspective to ABC, and T_c is perspective to the medial triangle of ABC.

  The appearance of (i, j, m, n) in the following list means that, if X_i, X_j are the given centers on the inconics, then the perspector (T_p, ABC) is X(m) and the perspector (T_c, medial-of-ABC) is X_n: (1, 2, 59459, 59460), (1, 6, 59461, 59462), (1, 7, 59463, 59464), (1, 8, 59465, 59466), (1, 55, 59467, 59468), (1, 56, 59469, 59470), (2, 6, 1509, 17045), (2, 7, 59471, 59472), (2, 8, 59473, 59474), (7, 8, 6063, 2886), (7, 55, 59475, 59476), (7, 56, 552, 59477), (8, 55, 261, 4999), (8, 56, 59478, 59479), (55, 56, 7, 1), (3, 4, 59482, 59483)^*

  ^(*): Only for ABC acute. For Brocard points PU(1), perspectors (T_p, ABC), (T_c, medial-of-ABC) are X₅₉₄₈₀, X₅₉₄₈₁, respectively. For isotomic conjugates of Brocard points, PU(11), perspectors (T_p, ABC), (T_c, medial-of-ABC) are X₄₀₀₉₉, X₂₆₅₅₈, respectively.

  A list of coordinates of A-vertices for each T_p, T_c treated in this section can be seen here (X(59459).pdf).

• X₅₉₈₂₉–X₅₉₉₉₃ — Centers of orthogonal circles

  contributed by César Eliud Lozada, October 22, 2023. .LI li{padding-bottom:12px;} Given a circle Ω and two points :g:`P <P>`, :g:`Q <Q>` not lying on a diameter of Ω, there exists an unique circle Ω^⟂ through :g:`P <P>` and :g:`Q <Q>` and orthogonal to Ω.

  This circle is denoted here Ω^⟂(P, Q). Its center is the intersection of the perpendicular bisectors of PP' and QQ', where P', Q' are the inverses of P, Q in Ω. Take in account that, when a point P approaches to Ω, the perpendicular bisector of P and P' approaches to the tangent to Ω at P.

  Given a circle Ω, a line 𝓁 not passing through the center of Ω, and a point :g:`P <P>` on 𝓁, there exists an unique circle Ω^⟂ orthogonal to Ω and tangent to 𝓁 at :g:`P <P*>`.

  This circle is denoted here Ω^⟂(𝓁, P). Its center is the intersection of the perpendicular to 𝓁 through P and the perpendicular bisector of P and its inverse P' in Ω.

Part 31

• X₆₀₃₄₃–X₆₀₃₅₂ — Points associated with hyperbolas

  contributed by Clark Kimberling and Peter Moses, November 3, 2023. Let H be a hyperbola. Let W be the center of H, and let L and L' be the asymptotes of H. There exists a unique hyperbola H' other than H that has center W and asymptotes L and L'. The hyperbola H' is the called the conjugate of H. If H is a circumhyperbola and L is given by

  u x + v y + w z = 0,

  then L' is given by

  v w x + w u y + u v z = 0.

  The hyperbola H is given by u(v - w)^2 y z + (cyclic) = 0, with center u(v^2 - w^2) : : and perspector u(v - w)^2 y z : : .

  The conjugate hyperbola H' is given by u(v+w)^2 y z + (cyclic) + 2 u v w (x^2 + y^2 + z^2) = 0, with center u(v^2 - w^2) : : and perspector

  u*(v - w)*(3*u*v + v^2 + u*w + 3*v*w)*(u*v + 3*u*w + 3*v*w + w^2) : : .

  Example 1. H = Kiepert hyperbola

  Center of H and H': X₁₁₅

  Asymptotes, L and L', are the lines u x + v y + w z = 0, where \(u:v:w\) = X₃₀₅₀₈ and \(u:v:w\) = X₃₀₅₀₉.

  Equation for H: (b^2 - c^2) b y + (cyclic) = 0

  Equation for H': 2*((a^2 - b^2)^3*x*y + (-a^2 + c^2)^3*x*z + (b^2 - c^2)^3*y*z) - (a^2 - b^2)*(a^2 - c^2)*(b^2 - c^2)*(x^2 + y^2 + z^2) = 0

  The point X_i lies on H' for these i: 3413, 3414, 39107, 39108. The perspector of H' is X₉₂₉₃.

  Example 2. H = Jerabek hyperbola

  Center of H and H': X₁₂₅

  Asymptotes, L and L', are the lines u x + v y + w z = 0, where \(u:v:w\) = X₅₀₉₄₄ and \(u:v:w\) = X₅₀₉₄₅.

  Equation for H: a^2(b^2 - c^2)SA y z + (cyclic) = 0

  Equation for H': 2*(a^2*b^2*(a^2 - b^2)^3*(-a^2 - b^2 + c^2)*x*y + a^2*c^2*(-a^2 + b^2 - c^2)*(-a^2 + c^2)^3*x*z + b^2*c^2*(a^2 - b^2 - c^2)*(b^2 - c^2)^3*y*z) - (a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(x^2 + y^2 + z^2) = 0

  The point X_i lies on H' for these i: 2574, 2575. The perspector of H' is X₆₀₄₇₈.

  Example 3. H = Feuerbach hyperbola

  Center of H and H': X₁₁

  Asymptotes, L and L', are the lines u x + v y + w z = 0, where \(u:v:w\) = X₆₀₄₇₆ and \(u:v:w\) = X₆₀₄₇₇.

  Equation for H: a(b - c)(b + c - a) y z + (cyclic) = 0

  Equation for H': 2*(a*(a - b)^3*b*(a + b - c)*x*y + a*c*(-a + c)^3*(a - b + c)*x*z + b*(b - c)^3*c*(-a + b + c)*y*z) + (a - b)*(b - c)*(a + b - c)*(-a + c)*(a - b + c)*(-a + b + c)*(x^2 + y^2 + z^2) = 0

  The point X_i passes through H' for these i: 3307, 3308. The perspector of H' is X₄₂₅₅₂.

  Starting with a line L, the L-asymptotic circumhyperbola is the hyperbola that passes through the vertices A, B, C and has L as an asymptote. (Further examples are omitted here.]

• X₆₀₃₅₃–X₆₀₄₇₅ — Common point of radical axes

  contributed by César Eliud Lozada, November 5, 2023. Let ω be a circle and P, Q two distinct fixed points, none on ω. Then the radical axes of ω and all the circles through :g:`P <P>` and :g:`Q <Q>` have a common point X(ω, P, Q).

  The pencil or set of circles through P, Q is denoted here OO(P, Q).

  Some properties: .LI li{padding-bottom:12px;} The common point X(ω, P, Q) lies on the line PQ. Therefore, it is the intersection of this line with the radical axis of ω and any chosen circle in O(P, Q). A very simple proof of this fact can be seen here (X60353.pdf). X(ω, P, Q) is the radical center of ω and any pair of circles in O(P,Q). When P, Q and the center of ω are not collinear, X(ω, P, Q) = PQ∩P'Q', where P' and Q' are the respective inverses of P, Q in ω. If P or Q is the center of ω then X(ω, P, Q) is the inverse of the other in ω. If ω is the circumcircle of ABC then X(ω, P, Q) concides with the Vu (P,Q)-circle point (see preamble just before X₃₈₄₅₈). If ω is the circumcircle of ABC and P or Q is the centroid X₂ of ABC, then X(ω, P, Q) coincides with the Vu pole of P and Q (see preamble just before X₃₇₇₅₆).

• X₆₀₅₃₀–X₆₀₅₅₁ — Dao-Lozada circum-bicevian-perspectors

  contributed by César Eliud Lozada, November 10, 2023. Let ABC be a triangle with circumcircle ω. Let P', P″ be two interior points to ABC and A'B'C', A″B″C″ their respective cevian triangles. Denote \(a_{t}\) the circle through A' and A″ tangent to ω at A_t, with A_t lying on the arc BC of ω not containing A. Define \(b_{t}\), B_t, \(c_{t}\), C_t cyclically. Then the lines AA_t, BB_t, CC_t concur at a point Q(P', P″). (Dao Thanh Oai, November 6, 2023).

  The point Q(P', P″) is named here the Dao-Lozada circum-bicevian-perspector of P’ and P”.

  If P' =\(x' : y' : z'\) and P″ = x” : y” : z” (barycentrics), then A_t = -a^2/(c*Y + b*Z) : b/Z : c/Y and Q(P', P″) = a*X : b*Y : c*Z where X = \(\sqrt{x'*x"}\), Y = \(\sqrt{y'*y"}\), Z = \(\sqrt{z'*z"}\).

  In general, the above concurrence does not occur for the second circles through the traces of P', P″, tangent to ω and touching it at points on its positive arcs.

  The appearance of (i, j, k) in the following list means that Q(X_i, X_j) = X_k: (1, 2, 365), (1, 6, 18753), (1, 7, 266), (1, 8, 259), (1, 9, 60530), (1, 10, 60531), (1, 11, 60532), (1, 12, 60533), (2, 6, 6), (2, 7, 509), [and others].

• X₆₀₅₅₂–X₆₀₅₆₄ — Circumtangential-bicevian-perspectors

  contributed by César Eliud Lozada, November 10, 2023. Continuing with the construction and notations in the previous section (see preamble just before X₆₀₅₃₀), let A*B*C* be the triangle bounded by the tangent lines to ω at A_t, B_t, C_t. Then A*B*C* is perspective to ABC with perspector Q*(P', P″).

  This new perspector is referred here as the circumtangential-bicevian-perspector of P’ and P”. Corresponding barycentrics coordinates are: A* = -a*(2*b*c*X^2 + a*(a*Y*Z + b*Z*X + c*X*Y)) : b^2*(a*Y*Z + b*Z*X - c*X*Y) : c^2*(a*Y*Z - b*Z*X + c*X*Y) and Q*(P', P″) = a^2/(-a*Y*Z + b*Z*X + c*X*Y) : b^2/(a*Y*Z - b*Z*X + c*X*Y) : c^2/(a*Y*Z + b*Z*X - c*X*Y)

  The circumtangential-bicevian-perspector of P' and P″ results to be the U-vertex conjugate of-U, where U is the Dao-Lozada-circum-bicevian perspector of P' and P″ explained in the previous section.

• X₆₀₆₀₃–X₆₀₆₁₁ — Points associated with the Neuberg-Gibert hyperbola

  contributed by Peter Moses, November 14, 2023, based on notes about “hyperbola (P)” in Bernard Gibert’s webpage, K001, the Neuberg cubic. Gibert’s notes include the following: (P) is a very remarkable hyperbola passing through X₄₇₆ and the vertices of the circumtangential triangle T_aT_bT_c. It has two asymptotes making an angle of 60 degrees, so that its eccentricity is 2. X₁₁₀ is one of its foci and the related directrix is the Euler line. The tangent at X₄₇₆ is the real asymptote of the Neuberg cubic. The hyperbola (P) is here named the Neuberg-Gibert hyperbola. Associated triangle centers include the following:

  X₆₀₆₀₃ = center

  X₆₀₆₀₄ = focus, other than X₁₁₀

  Pass-through points: X₄₇₆ and X_i for these i: 60605, 60606, 60607, 60608, 60609, 60610, 60611. The asmptotes meet the infinity line in PU(215).

• X₆₀₆₁₄–X₆₀₇₃₇ — Bicevian Chordal Triangles

  contributed by Ivan Pavlov on November 18, 2023. Let (c) be the bicevian conic of P=\(u:v:w\) and Q=\(p:q:r\) in barycentrics. Denote by Ap, Aq the intersection points of AP, AQ and (c), and similarly define Bp, Bq, Cp, and Cq. The lines ApAq, BpBq, and CpCq form a triangle A'B'C' which we call the bicevian chordal triangle of P and Q (wrt ABC).

  A'B'C' and ABC are perspective with center which lies on the circumconic through P and Q. We call this center the bicevian chordal perspector of P and Q (wrt ABC).

  A first barycentric coordinate is p*u*(r^2*u*v+2*r*(q*u+p*v)*w+p*q*w^2)*(p*r*v^2+q^2*u*w+2*q*v*(r*u+p*w))

  For details, see Euclid 6040.

• X₆₀₇₄₂–X₆₀₇₇₃ — Centers of coaxial circles

  contributed by César Eliud Lozada, November 25, 2023. Given two non-concentric circles 𝒞₁ and 𝒞₂ and a point P, neither on any of the given circles nor on their radical axis, there exists an unique circle through P and coaxial with the given circles. (A simple proof and a method for determining this circle can be seen here (X(60742).pdf).)

  Such circle is denoted here as Ω_x(𝒞₁, 𝒞₂, P).

• X₆₀₇₈₈–X₆₀₈₄₃ — Chordal perspectors of bicevian conics and pedal circles

  contributed by César Eliud Lozada, December 3, 2023. Let ABC be a triangle, P', P″ two distinc points, none on their sidelines, and A'B'C', A″B″C″ their respective cevian triangles with respect to ABC. Call 𝒞 the bicevian conic of P' and P″.

  Let P be a point on the line P'P″ and denote A₁, B₁, C₁ the second intersections of 𝒞 and the lines PA', PB', PC', respectively. Similarly, denote A₂, B₂, C₂ the second intersections of 𝒞 and the lines PA″, PB″, PC″, respectively. Then the lines AA₁, BB₁, CC₁ concur in a point Q₁ and the lines AA₂, BB₂, CC₂ concur in a point Q₂.

  The point Q₁ is denoted here the (P’, P”)-bicevian conic chordal perspector of-P whilst the point Q₂ is denoted as the (P”, P’)-bicevian conic chordal perspector of-P.

  ⬥ Pedal triangles version

  Let ABC be a triangle, P' a point not on their sidelines, P″ the isogonal conjugate of P' and A'B'C', A″B″C″ their respective pedal triangles with respect to ABC. Call 𝒞 the circle through A', B', C', A″, B″, C″.

  Let P be a point on the line P'P″ and denote A₁, B₁, C₁ the second intersections of 𝒞 and the lines PA', PB', PC', respectively. Similarly, denote A₂, B₂, C₂ the second intersections of 𝒞 and the lines PA″, PB″, PC″, respectively. Then the lines AA₁, BB₁, CC₁ concur in a point Q₁ and the lines AA₂, BB₂, CC₂ concur in a point Q₂.

  In this case, the point Q₁ is denoted here the P’-pedal circle chordal perspector of-P and, naturally, the point Q₂ is denoted as the P”-pedal circle chordal perspector of-P. In this notation, the term “pedal circle” may be replaced with the name of the circle, if it has a given name. Therefore, the (X₂, X₄)-bicevian conic chordal perspector of-P coincides with the X₃-nine-point circle chordal perspector of-P and the (X₄, X₂)-bicevian conic chordal perspector of-P coincides with the X₄-nine-point circle chordal perspector of-P.

• X₆₀₈₇₇–X₆₁₀₃₅ — Points related to the Aguilera triangle

  contributed by Ivan Pavlov on Dec 11, 2023. In a scalene acute triangle ABC, let M_aM_bM_c be its medial triangle. Let ωa be the circle centered at M_a and passing through B and C, and define ωb and ωc cyclically. Inside ABC, let O_a be center of the circle, closest to A, which is externally tangent to ωa and is inscribed in angle BAC. Define O_b and O_c cyclically. The triangle O_aO_bO_c is called the (1st) Aguilera triangle of ABC and has the following barycentrics of the A-vertex, derived by Manuel Aguilera: a*(-a + b + c)*(a + b + c) + 2*(b + c)*S : b*((-a + b + c)*(a + b + c) - 2*S) : c*((-a + b + c)*(a + b + c) - 2*S)

  If a point P lies on line X₁X₃, the pedal triangle of P is orthologic to the Aguilera triangle and the orthology center lies on line X₂X₇.

  The triangle inverse-in-incircle of the Aguilera triangle is called here the (1st) Aguilera-Pavlov triangle.

  Its A-vertex has the following barycentrics: S (b + c) + 2 a b c : b (-S + 2 b c ) : c (-S + 2 b c )

  The centroid of the Aguilera-Pavlov triangle coincides with the incenter of ABC.

  If a point P lies on line X₁X₆, the pedal triangle of P is orthologic to the Aguilera-Pavlov triangle.

• X₆₁₁₀₃–X₆₁₁₃₈ — Orthoptic or director circles

  :contributed by César Eliud Lozada, January 11, 2024. Let 𝒞 be a conic and let ℒ be the locus of points from which the tangent lines to 𝒞 are perpendicular. ℒ is, in general, a circle centered at the center of 𝒞 and named the orthoptic or director circle of 𝒞.

  If 𝒞 is a parabola then ℒ degenerates to the directrix of 𝒞, and, if 𝒞 is a rectangular hyperbola, ℒ degenerates to the center of 𝒞.

  When 𝒞 is an ellipse with semiaxes 𝒶 and 𝒷 then its orthoptic circle has squared-radius ρ² = 𝒶² + 𝒷². This means that, if 𝒞 is a circle with radius 𝓇, then its orthoptic circle has squared-radius ρ² = 2*𝓇².

  Finally, if 𝒞 is an hyperbola with semiaxes 𝒶 (focal) and 𝒷 then its orthoptic circle has squared-radius ρ² = 𝒶² - 𝒷². Therefore, the orthoptic circle exists only when 𝒶 ≥ 𝒷.

  In S.L. Loney, The Elements of Coordinate Geometry, 1962, pp 365, #390, a general expression is deduced for calculating the equation of the director or orthoptic circle of a conic (in cartesian coordinates). Such expression, when applied to the conic given in barycentric as 𝒞 = Σ(F_A*x^2 + 2*G_A*y*z) = 0, leads to the following equation for the squared-radius of the orthoptic circle:

  ρ^2 = (∑(F_A*G_A^2)-F_A*F_B*F_C-2*G_A*G_B*G_C)*∑((F_B+F_C-2*G_A)*SA)/∑(F_B*F_C-2*F_A*G_A-G_A^2+2*G_B*G_C)^2 (all sums are cyclic)

• X₆₁₁₄₆–X₆₁₁₅₁ — Points associated with circles

  contributed by Peter Moses and Clark Kimberling, January 16, 2024. Suppose that n >= 2 and that S = {O(1), O(2), … , O(n)} is a set of n circles with centers and radii o(1), r(1); o(2), r(2); …; o(n),r(n), where the centers o(i) are normalized barycentric coordinates.

  Definition 1. The centroid of S is the point o(1) + o(2) + … + o(n), this being a combo as defined in the Introduction (in Part 1 of ETC).

  Definition 2. The centroid of circumferences of S is the point r(1)*o(1) + r(2)*o(2) + … + r(n)*o(n).

  Definition 3. The centroid of curvatures of S is the point o(1)/r(1) + o(2)/r(2) + … + o(n)/r(n).

  Definition 4. The centroid of areas of S is the point o(1)*r(1)^2 + o(2)*r(2)^2 + … + o(n)*r(n)^2.

  . Definition 5. The centroid of reciprocal areas of S is the point o(1)/r(1)^2 + o(2)/r(2)^2 + … + o(n)/r(n)^2.

  All five centroids are given by the form o(1)*r(1)^n + o(2)*r(2)^n) + … + + o(n)*r(n)^n, where n is one of the numbers -2, -1, 0, 1, 2. In the following examples, the centroids are indexed by n, from -2, to 2, with these designations: G(-2), G(-1), G(0), G(1), G(2).

  Examp1e 1: S = {incircle, circumcircle}

  G(-2) = X₈₀₇₁

  G(-1) = X₅₅

  G(0) = X₁₃₈₅ G(1) = X₆₁₁₄₆

  G(2) = X₆₁₁₄₇

  [Further examples are omittted here.]

• X₆₁₁₅₂–X₆₁₂₉₇ — and X₆₁₂₄₄-X₆₁₂₉₇, Miyamoto Perspectors

  submitted by Clark Kimberling, January 20, 2024, based on notes from Keita Miyamoto . Barycentrics for Miyamoto perspectors were found by Peter Moses. Let A'B'C' be a triangle homothetic to ABC at X₂ with ratio k. Let A_b=AB∩B'C', and define B_c and C_a cyclically. Let A_c=CA∩B'C', and define B_a and C_b cyclically. Let (I), (I_a), (I_b), (I_c) be the incircles of ABC, A'B_cC_b, B'C_aA_c, C'A_bB_a, respectively. Then there exists a circle (O(k)) tangent to all four circles, (I), (I_a), (I_b), (I_c). The touchpoint of (I) and (O(k)) is the Feuerbach point, X₁₁. Further, let T_a be the touchpoint of (I_a) and (O(k)), and define T_b and T_c cyclically. The lines AT_a, BT_b, CT_c concur in a point here named the Miyamoto (k)-perspector.

  Likewise, if A'B'C' is homothetic to an arbitrary triangle T = A″B″C″, X₂ with ratio k, then the above construction yields a point here named named the (T,k)-Miyamoto perspector. Fifty-four (Euler triangle, k)-Miyamoto perspectors, found by Peter Moses, are indexed at X₆₁₂₄₄-X₆₁₂₉₇.

  The points X₆₁₁₅₂-X(61159 lie on the line X₂X₁₁, and X₂₁₂₄₄-X₆₁₂₉₇ lie on X₁X₅.

  The appearance of (k,X_i) in the following list means that X_i = Miyamoto (k)-perspector.

  (-3,61152), (-2,61153), (3,61154), (1/2,61155), (-3/2,61156), (3/2,61157), (-2/3, 61158), (2/3,61159)

  (-8,61244), (-13/2,61245), (-17/4,61246), (-4,61247), (-16/5,61248), (-11/4,61249), (-13/5,61250), (-5/2,61251), (-17/7,61252), (-19/8,61253), (-5,3,61254), (-13/8,61255), (-11/7,61256), (-4/3,61257), (-8/7,61258), (-7/8,61259)

  (-5/6,61260), (-4/5,61261), (-3/4,61262), (-2/3,61263), (-3/5,61264), (-3/7,61265), (-2/5,61266), (-3/8,61267), (-2/7,61268), (-1/4,61269),

  (-1/6,61270), (-1/7,61271), (-1/8,61272), (1/6,61273), (1/5,61274), (1/3,61275), (2/5,61276), (4/7,61277), (5/8,61278), (2/3,61279),

  (3/4,61280), (11/8,61281), (10/7,61282), (3/2,61283), (8/5,61284), (5/3,61285), (7/4,61286), (2,61287), (11/5,61288), (19/7,61289),

  (3/4,61290), (11/8,61291), (10/7,61292), (3/2,61293), (8/5,61294), (5/3,61295), (7/4,61296)

• X₆₁₃₀₁–X₆₁₄₁₈ — Vertex Square Sum and Product

  contributed by Ivan Pavlov on Jan 31, 2024. Given a reference triangle ABC, for any central triangle XYZ the barycentric sum X^2+Y^2+Z^2 is a triangle center. We call this expression the vertex square sum of XYZ. A curious example is that the vertex square sum of the anticevian triangle of a point P is P^2. Some other examples: The vertex square sum of the intouch triangle is X₂₃₆₅₃. The vertex square sum of the tangential triangle is X₃₂. The vertex square sum of the 1st Brocard triangle is X₂. The vertex square sum of the Moses-Soddy triangle is X₁₀₈₆. The vertex square sum of the Soddy triangle is X₂₇₉. The vertex square sum of the Pelletier triangle is X₁₄₉₃₆. The vertex square sum of the Schroeter triangle is X₁₁₅. We call the barycentric product X*Y*Z the vertex product of XYZ. Note that the vertex product of the anticevian triangle of a point P is P^3.

• X₆₁₇₄₇–X₆₁₇₆₁ — Reflected-parallels circles

  contributed by César Eliud Lozada, February 27, 2024. The following problem by C. Pohoata is published in AOPS: Three parallel lines \(p_{a}\), \(p_{b}\), \(p_{c}\) pass through the vertices of a triangle ABC. Their reflections in BC, CA, AB, respectively, form a triangle A'B'C'. Find the locus of the incenters of such triangles.

  When ABC is acute, the required locus is a circle centered at the circumcenter X₃-of-ABC and having radius ρ = 2*R. But, as a matter of fact, similar constructions with centers X_n, for 1 ≤ n ≤ 1000, result each in a circle as locus (conjectured and not formally proven yet, but numerically tested). The appearance of (i, j) in the following list means that the locus of centers X_i-of-A'B'C' is a circle with center X_j wrt ABC: (1, 3), (2, 61747), (3, 156), (4, 9927), (5, 13406), (6, 61748), (7, 34507), (8, 22802), (9, 34117), (10, 61749), (11, 5), (12, 61750), (20, 61751), (21, 10274), (35, 1614), (36, 110), (40, 32139), (46, 11441), (55, 61752), (56, 61753), (57, 15068), (65, 5876), (79, 2888), (80, 4), (84, 58726), (90, 2904), (100, 6759), (101, 61754), (104, 1147), (105, 61755), (108, 61756), (109, 61757), (110, 61758), (113, 61759), (117, 61760), (118, 61761), (119, 15761), (149, 18381), (177, 5694), (191, 17824), (214, 10282), (238, 1576), (354, 15067), (355, 44279), (496, 49673), (551, 10182), (942, 11591), (946, 5449), (954, 19127), (960, 41589)

  The locus of X_i-of-A'B'C' is denoted here as the reflected-parallels circle of X_i. It must be taken in account that the given results are valid as long as ABC is acute.

Part 32

• X₆₂₂₄₉–X₆₂₂₆₅ — Tran-Lozada perspectors

  contributed by César Eliud Lozada, March 16, 2024. The following two assertions, slightly modified, are proposed by Tran Viet Hung in Romantics of geometry, March 15, 2023:

  1. Let ABC be a triangle, P', P″ two distinct points and A'B'C', A″B″C″ their respective circumcevian triangles, such that A', A″ are in the same side with respect to the line BC, and similarly B', B″ and C', C″. Let (a*) be the circle through A' and A″ tangent to sideline BC, with center closer to the line BC. Let A_t be the touchpoint of (a*) and BC, and define B_t, C_t cyclically. Then the lines AA_t, BB_t, CC_t are concurrent in a point Q₁(P', P″).

  The point of concurrence Q₁(P', P″) is named here the Tran-Lozada bi-circumcevian perspector of P and P’. If P' =\(x' : y' : z'\) and P″ = x” : y” : z” (barycentrics), then Q₁(P', P″) = \(\sqrt{x' x"}\)/a : \(\sqrt{y' y"}\)/b : \(\sqrt{z' z"}\)/c. From here, it is clear than P', P″ must be both interior to ABC in order Q₁(P', P″) to be real. The barycentric coordinates of the center A* of (a*) are:

  A* = a^2*(2*\(\sqrt{y'*y"*z'*z"}\)*SA-(y’*z”+y”*z’)*b*c) : (2*S^2*c^2*y’*y”+((y’*z”+y”*z’)*c*SC+2*\(\sqrt{y'*y"*z'*z"}\)*b*SB)*b^3)/b^2 : (2*S^2*b^2*z’*z”+((y’*z”+y”*z’)*b*SB+2*\(\sqrt{y'*y"*z'*z"}\)*c*SC)*c^3)/c^2

  The appearance of (i, j, k) in the folowing list means that Q₁(X_i, X_j) = X_k:

  (1, 2, 18297), (1, 6, 366), (1, 31, 1), (1, 32, 365), (1, 75, 75), (1, 76, 62249), [and others].

  2. Let ABC be a triangle with circumcircle (O), P', P_o two points, A'B'C' the cevian triangle of P' and A_oB_oC_o the circumcevian triangle of P_o. Let (a*) be the circle through A_o and tangent to sideline BC at A'. Let A″ be the second intersection of (O) and (a*) and build B″, C″ cyclically. Then the lines AA″, BB″, CC″ are concurrent in a point Q₂(P', P_o).

  Q₂(P', P_o) is named here the Tran-Lozada perspector of cevian-of-P' and circumcevian-of-P_o. If P' =\(x' : y' : z'\) and P_o =\(x_{o} : y_{o} : z_{o}\) (barycentrics), then Q₂(P', P_o) = (a*x’)^2/\(x_{o}\) : (b*y’)^2/\(y_{o}\) : (c*z’)^2/\(z_{o}\).

  The appearance of (i, j, k) in the folowing list means that Q₂(X_i, X_j) = X_k, for (i, j) ≤ 8:

  (1, 1, 31), (1, 2, 32), (1, 3, 25), (1, 4, 184), (1, 5, 54034), (1, 6, 6), (1, 7, 2175), (1, 8, 1397), (2, 1, 1), (2, 2, 6), (2, 3, 4), (2, 4, 3), (2, 5, 54), (2, 6, 2), (2, 7, 55), (2, 8, 56), [and others].

• X₆₂₂₆₆–X₆₂₂₇₈ — Tran-Lozada CCO- and OOC- perspectors

  contributed by César Eliud Lozada, March 18, 2024. The following two assertions, slightly modified, are proposed by Tran Viet Hung in Romantics of geometry, March 17, 2023:

  1. Let ABC be a triangle with circumcircle (O), P', P″, P_o three points, with P' ≠ P″, and A'B'C', A″B″C″ the cevian triangles of P' and P″, respectively, and A_oB_oC_o the circumcevian triangle of P_o. Let A_s be the second intersection of circles (O) and (A'A″A_o), and build B_s, C_s cyclically. Then the lines AA_s, BB_s, CC_s are concurrent in a point Q₁(P', P″; P_o).

  The point of concurrence Q₁(P', P″; P_o) is named here the Tran-Lozada CCO-perspector of (P',P″; P_o). If P' =\(x' : y' : z'\), P″ = x” : y” : z” and P_o =\(x_{o}: y_{o} : z_{o}\) (barycentrics), then Q₁(P',P″; P_o) = a^2 x’ x”/\(x_{o}\) : b^2 y’ y”/\(y_{o}\) : c^2 z’ z”/\(z_{o}\).

  The appearance of (i, j, k, n) in the folowing list means that Q₁(X_i, X_j; X_k ) = X_n, for (i, j, k) ≤ 6:

  (1, 2, 1, 6), (1, 3, 1, 184), (1, 4, 1, 25), (1, 5, 1, 51), (1, 6, 1, 32), [and others].

  2. Let ABC be a triangle, P'_o, P″_o, P_i three points with P'_o ≠ P″_o, A'_oB'_oC'_o, A″_oB″_oC″_o the circumcevian triangles of P'_o and P″_o, respectively, and A_iB_iC_i the cevian triangle of P_i. Let A_s be the second intersection of the line BC and the circle (A'_oA”_oA_i) and build B_s, C_s cyclically. Then the lines AA_s, BB_s, CC_s are concurrent in a point Q₂(P'_o, P″_o; P_i).

  Q₂(P'_o, P″_o; P_i) is named here the Tran-Lozada OOC-perspector of (P'_o, P″_o; P_i). If P'_o =\(x'_{o} : y'_{o} : z'_{o}\), P″_o = x”_o : y”_o : z”_o, P_i =\(x_{i} : y_{i} : z_{i}\) (barycentrics), then Q₂(P'_o, P″_o; P*) = x’_o x”_o/(a^2* \(x_{i})\) : y’_o y”_o/(b^2 \(y_{i})\) : z’_o z”_o/(c^2 \(z_{i})\).

  The appearance of (i, j, k, n) in the folowing list means that Q₂(X_i, X_j; X_k ) = X_n, for (i, j, k) ≤ 6:

  (1, 2, 1, 76), (1, 3, 1, 69), (1, 4, 1, 264), (1, 5, 1, 311), (1, 6, 1, 2), (2, 3, 1, 304), (2, 4, 1, 1969), [and others].

  Note added by César E. Lozada on March 10, 2025:

  The steps for finding CCO(U, V; W) and OOC(U, V; W) allow the geometric construction of points algebraically defined, as the barycentric and trilinear product or quotient of two points.

  Suppose that two points U₁ and U₂ are given. It can be easily proved that: .LI li{padding-bottom:8px; before.content: counter(list, decimal) “) “; } barycentric product U₁*U₂ = CCO(U₁, U₂; X₆ ) = OOC(U₁, U₂; X₇₆ ) barycentric quotient U₁/U₂ = CCO(X₇₆, U₁; U₂ ) = OOC(X₆, U₁; U₂) trilinear product U₁*U₂ = CCO(U₁, U₂; X₃₁ ) = OOC(U₁, U₂; X₇₅) trilinear quotient U₁/U₂ = CCO(X₇₅, U₁; U₂ ) = OOC(X₃₁, U₁; U₂)

• X₆₂₂₈₀–X₆₂₂₈₆ — Seven-circles points

  contributed by César Eliud Lozada, March 21, 2024.The following theorem appears in C. J. A. Evelyn, G. B. Money-Coutts, J. A. Tyrrell, The seven circles theorem and other new theorems, Great Britain, 1974:

  Let (A'), (B'), (C') be three circles externally tangent (or internally tangent) to a circle (O), and touching this at A₁, B₁, C₁, respectively. Let (A″) be the circle externally tangent to (B'), (C'), and externally tangent to (O) (or internally tangent to (O), but with center A″ in the side of B'C' not containing A'), touching (O) at A₂, and define (B″), (C″), B₂, C₂ cyclically. Then, whenever the seven circles can be built, the lines A₁A₂, B₁B₂, C₁C₂ concur. (See figure here (.\X(62280).jpg))

  The points of concurrence Q_e (or Q_i) for both cases are denoted here as the external (or internal) seven circles point of circles (A’), (B’), (C’).

  The appearance of (Γ, i, j) in the following list means that the external- and internal- seven circles points of circles Γ are X_i and X_j, respectively:

  (excircles, 62280, 62281), (excosine, 6221, 6398), [and others].

• X₆₂₄₃₉–X₆₂₄₈₈ — Centers related to PU(217)-PU(237)

  contributed by César Eliud Lozada, April 5, 2024.

• X₆₂₄₈₉–X₆₂₅₁₀ — Infinity bisectors

  contributed by Clark Kimberling and Peter Moses, April 5, 2024. Let O denote the circumcenter, (O) the circumcircle, and L the line at infinity. Suppose that P =\(p:q:r\) and U =\(u:v:w\) are points on (O) and that P, O, U are noncollinear. Let L₁ be the tangent to (O) at P and L₂ the tangent to (O) at U. Let D = L₁∩L₂ and M = OD∩L. As the line OM bisects the angle between L₁ and L₂, the point M is here named the (P,U)-infinity bisector. Barycentrics for the (P,U)-infinity bisector are given by

  (\(a^{2}\) - \(b^{2}\) + \(c^{2})\)(q u - p v) - (\(a^{2}\) + \(b^{2}\) - \(c^{2})\)(r u - p w) - \(2a^{2}(r v - q w)\) : :

  The appearance of {{i,j},k} in the following lists means that X_k = {X_i,X_j}-infinity bisector.

  {{74,98},542}, {{74,99},690}, {{74,100},8674}, {{74,101},2774}, {{74,102},2779}, {{74,103},2772}, {{74,104},2771}, {{74,105},2836}, {{74,106},2842}, {{74,107},9033}, {{74,108},2850}, {{74,109},2773}, {{74,110},526}, {{74,111},2854}, {{74,112},9517}, {{74,476},523}, {{74,477},30},

  {{98,99},804}, {{98,100},2787}, {{98,101},2786}, {{98,102},2792}, {{98,103},2784}, {{98,104},2783}, {{98,105},2795}, {{98,106},2796}, {{98,107},2797}, {{98,108},2798}, {{98,109},2785}, {{98,110},690}, {{98,111},543}, {{98,112},2799}, {{98,476},62489}, {{98,477},62490}, [and many others].

  The line PU is the polar of D with respect to the circumcircle and OD is perpendicular to PU. Then M, the infinite bisector of {P,U}, is the orthopoint of the point at infinity of the line PU. (César Lozada, April 7, 2024)

• X₆₂₅₁₁–X₆₂₅₂₉ — Centers related to PU(202)-PU(212)

  contributed by César Eliud Lozada, April 7, 2024.

• X₆₂₅₀₀–X₆₃₂₂₄ — and X₆₂₇₂₁-X₆₂₇₃₃ and X₆₃₂₁₆-X₆₃₂₂₄, Centers associated with unary operations

  contributed by Clark Kimberling and Peter Moses, April 2024.

  A unary operation on homogeneous coordinates \(x : y : z\) (barycentric or trilinear) is a mapping that takes the point \(x:y:z\) to the point f(\(x:y:z\)} : f(y,z,x) : f(z,x,y) for some homoeneous function f. Introduced here are several examples:

  u₁(\(x:y:z\)) = (y-z)/x : (z-x)/y : (x-y)/z

  u₂(\(x:y:z\)) = x/(y-z) : y/(z-x) : z/(x-y)

  u₃(\(x:y:z\)) = (-2x+y+z)/x : (-2y+z+x)/y : (-2z+x+y)/z

  u₄(\(x:y:z\)) = x/(-2x+y+z) : y/(-2y+z+x) : z/(-2z+x+y)

  u₅(\(x:y:z\)) = (y-z)/(y+z) : (z-x)/(z+x) : (x-y)/(x+y)

  u₆(\(x:y:z\)) = (y+z)/(y-z) : (z+x)/(z-x) : (x+y)/(x-y)

  u₇(\(x:y:z\)) = (-2x+y+z)/(y+z) : (-2y+z+x)/(z+x) : (-2z+x+y)/(x+y)

  u₈(\(x:y:z\)) = (y+z)/(-2x+y+z) : (z+x)/(-2y+z+x) : (x+y)/(-2z+x+y)

  u₉(\(x:y:z\)) = (yz-zx-xy)/(y^2-z^2) : (zx-xy-yz)/(z^2-x^2) : (xy-yz-zx)/(x^2-y^2)

  u₁₀(\(x:y:z\)) = (y^2-z^2)/(yz-zx-xy) : (z^2-x^2)/(zx-xy-yz) : (x^2-y^2)/(xy-yz-zx)

  u₁₁(\(x:y:z\)) = (yz-zx-xy)/(y^2+z^2) : (zx-xy-yz)/(z^2+x^2) : (xy-yz-zx)/(x^2+y^2)

  u₁₂(\(x:y:z\)) = (y^2+z^2)/(yz-zx-xy) : (z^2+x^2)/(zx-xy-yz) : (x^2y^2)/(xy-yz-zx)

  In that list above, the 12 unary operations are indexed so that for n = 1,2,3,4,5,6, \(u_{2n}(X)\) is the isotomic conjugate of \(u_{2n-1}(X)\) when the coordinates are barycentric, and the isogonal conjugate when the coordinates are trilinear. In addition to the notations “u_n(\(x:y:z\))” and “\(u_{n}(X)\)” the notation “unary(n) of X″ will be useful. In the naming of triangle centers “unary(n) of X″ is used when the underlying coordinates are barycentric, and “trilinear unary(n) of X″ when the coordinates of trilinear. For examples of such points, see X(63202-X₆₃₂₀₅ and X₆₂₇₃₄-X₆₂₇₅₀.

  In the next table, column 1 represents the triangle centers X₁, X₃, X₄, …, X₁₁. The appearance of k in (row r, column n) means that \(u_{r}\)(X_n) = X_k. In this table, it is assumed that the coordinates used to define the unary operations are barycentric coordinates. [Table omitted here.)

  In the next table, column 1 represents the triangle centers X₂, X₃, X₄, …, X₁₁. The appearance of k in (row r, column n) means that \(u_{r}\)(X_n) = X_k. Here it is assumed that the coordinates used to define the unary operations are trilinear coordinates. [Table omitted here.]

• X₆₂₇₇₀–X₆₃₁₉₅ — Perspectors related to PTC triangles

  contributed by Ivan Pavlov on April 26, 2024. For a triangle ABC, and arbitrary points P, Q, and R not on its sides, let A' be the intersection of AP and the perpendicular through Q to BC and similarly define B' and C'. Let A″ be the intersection of RA' and BC and similarly define B″ and C″. Below, we denote with PTC(P,Q,R) the triangle A″B″C″. If P, Q, and R are triangle centers then PTC(P,Q,R) is a central triangle.

  If in barycentrics P=\(u:v:w\), Q=\(p:q:r\), and R=\(l:m:n\) then the A-vertex of PTC(P,Q,R) is

  0 : (b^2 - c^2) (-l (q + r) v + m p (v + w)) - a^2 ((l (-q + r) + m (p + 2 r)) v - m (p + 2 q) w) : (b^2 - c^2) (-l (q + r) w + n p (v + w)) - a^2 (n (p + 2 r) v + (-n (p + 2 q) + l (-q + r)) w)

• X₆₃₂₅₄–X₆₃₄₅₆ — Points related to the 1st and 2nd Pavlov triangles

  contributed by Ivan Pavlov on May 16, 2024. Let I_aI_bI_c be the cevian triangle of the incenter I for any triangle ABC. Denote with A_b the projection of I_a upon BI, and with B_a the projection of I_b upon AI.Similarly define B_c, C_b, A_c, and C_a. Lines A_bB_a, B_cC_b, and A_cC_a form a triangle A₁B₁C₁, here named the 1st Pavlov triangle. Lines A_bA_c, B_a B_C, and C_aC_b form a triangle A₂B₂C₂, here named the 2nd Pavlov triangle.

  For more infomation see Euclud 6171.

• X₆₃₄₇₈–X₆₃₅₂₈ — Bicevian centroidal collineation images

  , contributed by Ivan Pavlov on May 21, 2024. We will call {P,Q}-bicevian centroidal collineation the collineation which takes the cevian triangle of P = (\(u:v:w\)) into the cevian triangle of Q=(\(p:q:r\)) and has a fixed point in the centroid G of the reference triangle.

  The general formula for the image of X=(\(x:y:z\)) in barycentrics is

  𝓒𝓛(X,P,Q; G) = ( p (u (v + w) - v w) (v w (p u (q - r) (v - w) - q r v w) x + u (w y - v z) (p v w (q - r) - q r u (v - w))) : : )

  This transformation has the obvious inverse: 𝓒𝓛(X,P,Q; G)^-1 = 𝓒𝓛(X,Q,P; G).

  Some existing triangle centers are presented in the table below: [Table omitted here].

• X₆₃₆₄₀–X₆₃₆₅₅ — Hatzipolakis-Euler images

  contributed by César Eliud Lozada, May 28, 2024. Let ABC be a triangle, P a point and Q a point on its Euler line. Let Q_a, Q_b, Q_c the same to Q points on the Euler lines of the triangles PBC, PCA, PAB, respectively. Let \(p_{a}\), \(p_{b}\), \(p_{c}\) be parallel lines to PA, PB, PC through Q_a, Q_b, Q_c, respectively. What is the locus of P such that \(p_{a}\), \(p_{b}\), \(p_{c}\) concur?. (Antreas Hatzipolakis, euclid 6199.)

  If Q is such that O_ΔQ_Δ/O_ΔH_Δ = t = constant number, for a triangle Δ, then the lines \(p_{a}\), \(p_{b}\), \(p_{c}\) concur for every P. (César Lozada, euclid 6200.)

  The point of intersection of the given three lines is named here the Q-Hatzipolakis-Euler image of P.

  Note that the Dao image of P, defined in the preamble just before X₁₅₃₄₅, is the X₃-Hatzipolakis-Euler image of P. In general, if P_D is the Dao image of P, G is the centroid X₂ of ABC, and Z is the Q-Hatzipolakis-Euler of P, then GZ=(1-3*t)*GP_D. From this, it is clear that, for t=1/3, i.e., for Q=X₂, the X₂-Hatzipolakis-Euler image of P is X₂, for every P.

  Another interesting property of this image is that (Q-Hatzipolakis-Euler image of X₁₃) = (Q-Hatzipolakis-Euler image of X₁₄) = X₂, for every Q.

  Some Q-Hatzipolakis-Euler images of P are showed in the following table: [table omitted here].

• X₆₃₆₅₆–X₆₃₇₄₀ — Points related to the 2nd Hatzipolakis-Moses triangle

  contributed by Ivan Pavlov on May 29, 2024. Let ABC be a triangle and A'B'C' the orthic triangle. Denote Nab, Nac = the NPC centers of AHB', AHC'. Similarly define Nbc, Nba and Nca, Ncb.

  Let A″B″C″= triangle bounded by NabNac NbcNba, NcaNcb. It can be shown that A'B'C' and A″B″C″ are homothetic and this consutrction can be generalized for any pedal triangle.

  For more information see this Euclid thread.
  Below we call A″B″C″ the 2nd Hatzipolakis-Moses triangle. The barycentric coordinates of its A-vertex are: (

  a^2 (-2 a^2 (b^2 - c^2)^2 + a^4 (b^2 + c^2) + (b^2 - c^2)^2 (b^2 + c^2)) :

  -2 a^8 + (b^2 - c^2)^3 (b^2 + c^2) - a^4 (b^2 + 3 c^2)^2 + a^6 (4 b^2 + 7 c^2) - a^2 (2 b^6 + 3 b^4 c^2 - 5 c^6) :

  -2 a^8 - (b^2 - c^2)^3 (b^2 + c^2) - a^4 (3 b^2 + c^2)^2 + a^6 (7 b^2 + 4 c^2) - a^2 (-5 b^6 + 3 b^2 c^4 + 2 c^6))

  Note that the unary cofactor triangle of a given triangle, which is used in the definition of some centers is defined here: https://mathworld.wolfram.com/UnaryCofactorTriangle.html

• X₆₃₇₄₁–X₆₃₇₄₅ — Perspectors associated with triangles T(u_1,P*)

  contributed by Clark Kimberling and Peter Moses, May 29, 2024. Let P =\(p : q : r\) be a triangle center and

  A' = (q - r)/(q + r) : - (q + 2r)/q : (2q + r)/r

  B' = (2r + p)/p : (r - p)/(r + p) : - (r + 2p)/r

  C' = - (p + 2q)/p : (2p + q)/q : (p - q)/(p + q)
  The triangle A'B'C' is here denoted by T(u_1, P*)

  If P lies on the Steiner circumellipse, then A'B'C' is perspective to ABC, and the perspector is the barycentric quotient U/P, where U = reflection of P in X₂. This perspector lies on the Tucker nodal cubic, K015. The appearance of (i,j) in the following list means that X_i is on the Steiner circumellipse, and the X_j is the perspector of ABC and T(u_1, X_i*).

  (99,5466), (190,6548), (648,34767), (664,63743), (668,43928), (670,63744), (671,5468), (892,34763), (903,17780), (1121,56543), (1494,4240), (2966,34765), (3227,41314), (3228,63742), (4555,34764), (4562,47070), (5641,34761), (6189,30508), (6190,30509), (16077,47071), (18823,34760), (32041,63221), (35153,34766), (35168,34762)

• X₆₃₇₄₆–X₆₃₇₄₉ — Perspectors associated with triangles T(u_2,P*)

  contributed by Clark Kimberling and Peter Moses, May 29, 2024. Let P =\(p : q : r\) be a triangle center and

  A' = (q + r)/(q - r) : - q/(q + 2r) : r/(2q + r)

  B' = p/(2r + p) : (r + p)/(r - p) : -r/(r + 2p)

  C' = -p/(p + 2q) : q/(2p + q) : (p + q)/(p - q)

  The triangle A'B'C' is here denoted by T(u_2, P*). If P lies on the Steiner circumellipse, then A'B'C' is perspective to ABC, and the perspector is the barycentric quotient P/U, where U = reflection of P in X₂. This perspector lies on the Tucker nodal cubic, K015. The appearance of (i,j) in the following list means that X_i is on the Steiner circumellipse, and the X_j is the perspector of ABC and T(u_2, X_i*).

  (99,5468), (190,17780), (290, 63746), (648,4240), (664,56543), (668,41314), (670,63747), (671,5466), (892,34760), (903,6548), (1121,63748), (1494,34767), (2481,63221), (2966,34761), (3227,43928), (4555,34762), (5641,34765), (6189,30509), (6190,30508), (18822,47070), (18823,34763), (3228,63749), (35148,34766), (35168,34764), (53201,47071)

• X₆₃₇₈₇–X₆₃₇₉₄ — Dao-Zeeman perspectors

  :contributed by César Eliud Lozada, June 9, 2024. Let ABC be a triangle, U a point on its plane, 𝓁 the tripolar of U with respect to ABC and P, Q any two distinct points on 𝓁.

  Let A', B', C' be the intersections of 𝓁 with BC, CA, AB, respectively.

  Denote by A_P the intersection of the parallel line to CP through B' and the parallel line to BP through C', and define B_P, C_P cyclically.

  Denote by A_Q the intersection of the parallel line to CQ through B' and the parallel line to BQ through C', and define B_Q, C_Q cyclically.

  Then the triangle A″B″C″ bounded by the lines A_PA_Q, B_PB_Q, C_PC_Q is congruent and homothetic with ABC and the homothetic center H(P, Q) lies on the line 𝓁.

  When PQ is the Euler line this result is Gossard perspector theorem.(See here) When PQ parallel to the Euler line, this result due to Zeeman generalization. (See here) When P is the centroid this result is the Yiu’s generalization. Dao Thanh Oai - June 8, 2024.

  Notes added by César Eliud Lozada:

  The points A″, B″, C″, H(P, Q) are independent of P and Q and depend only on U or, still better, on 𝓁. Therefore, H(P, Q) can be more simply expressed as H(U) or H(𝓁), this last form preferred here. H(𝓁) is referred here as the Dao-Zeeman perspector of the line 𝓁.

  If U =\(x:y:z\) (barycentrics) then \(H(U) = x (y-z) \cdot (x-y-z)\): :. It can be deduced that, algebraically, Q(U) = Complement(IsotomicConjugate(Cevapoint(U, IdealPointOfTripolar(U)))).

  The appearance of (𝓁, n) in the following list means that H(𝓁) = X_n: (antiorthic axis X₄₄X₅₁₃, 650), (Brocard axis X₃X₆, 34349), (Brocard line X₃₉X₅₁₂, 63787), (De Longchamps line X₃₂₅X₅₂₃, 23301), (Euler line X₂X₃, 402), (Fermat axis X₆X₁₃, 63788), (Gergonne line X₂₄₁X₅₁₄, 7658), (IO line X₁X₃, 34345), (Lemoine axis X₁₈₇X₂₃₇, 647), (Nagel line X₁X₂, 62630), (Napoleon axis X₆X₇, 63789), (orthic axis X₂₃₀X₂₃₁, 6587), (Sherman line X₃₂₅₉X₃₃₂₆, 45950), (Soddy line X₁X₇, 63790), (van Aubel line X₄X₆, 63791), (X₁X₄, 63792), (X₁X₆, 63793), (X₂X₆, 11053), (X₄X₆, 63794)

  The appearance of (i, j) in the following list means that H(X_i) = X_j:

  (1, 650), (3, 647), (4, 6587), (5, 17434), (6, 647), (7, 7658), (8, 4521), (9, 650), (10, 661), (11, 17435), (25, 52588), (30, 14401), (37, 661), (39, 3005), (42, 52592), (57, 6129), (75, 3835), [and others].

• X₆₃₈₀₁–X₆₃₈₀₉ — Trigonometric Sums

  contributed by Clark Kimbering and Peter Moses, June 11, 2024. This section treats triangle centers of the forms sin(nB+mC)+sin(mB+nC) and cos(nB+mC)+cos(mB+nC). The appearance of (n,m,k) in the following list means that X_k = sin(nB+mC)+sin(mB+nC):

  (5,3,63801), (4,4,1147), (4,2,1154), (4,0,5449), (4,-2,565), (3,3,6149), (3,2,63801),(3,1,44706),(3,0,63803), (3,-1,564), (2,2,3), (2,0,5),(2,1,63804) (1,1,1), (1,0,10).

  The appearance of (n,m,k) in the next list means that X_k = cos(nB+mC)+cos(mB+nC):

  (4,4,63762), (4,2,63805), (4,0,63806), (3,3,63760), (3,2,73807), (3,1,63808), (2,2,1993), (2,1,16577), (2,0,343), (2,-1,63809), (2,-2,63763), (1,1,63), (1,0,226), (1,-1,14213).

• X₆₃₈₂₅–X₆₃₈₃₇ — Trigonometric Differences and Products

  contributed by Clark Kimbering and Peter Moses, June 14, 2024. This section treats triangle centers of the forms sin(nB+mC)-sin(mB+nC) and cos(nB+mC)-cos(mB+nC) and also of the forms sin(nB+mC)*sin(mB+nC) and cos(nB+mC)*cos(mB+nC), etc.

  The appearance of (n,m,k) in the following list means that X_k = sin(nB+mC)-sin(mB+nC):

  (1,-1,1577), (1,0,514), (2,-2,18314), (2,0,525), (2,-1,63825), (2,1,14838), (3,1,63827), (3,2,63828), (3,3,63826), (4,2,63830), (4,4,63829)

  The appearance of (n,m,k) in the next list means that X_k = cos(nB+mC)-cos(mB+nC):

  (1,0,522), (2,0,523), (2,1,3738), (3,-1,2618), (3,1,656), (3,2,63831), (4,0,6368), (4,2,526), (5,3,63832)

  The appearance of (n,m,k) in the next list means that X_k = sin(nB+mC)*sin(mB+nC):

  (1,-1,338), (1,0,75), (1,1,6), (2,0,264), (2,1,662), (2,2,577), (3,0,63759), (3,1,63833), (3,3,63834), (4,0,55553), (4,2,18315)

  The appearance of (n,m,k) in the next list means that X_k = cos(nB+mC)*cos(mB+nC):

  (1,-1,45793), (1,0,92), (1,1,394), (2,0,5392), (2,1,2167), (2,2,63835), (3,0,63764), (3,1,63836), (4,0,63765), (4,2,63766)

  The appearance of (n,m,k) in the next list means that X_k = cot(nB+mC) + cot(mB+nC):

  (1,0,6}, (1,1,69}, (2,0,577}, (2,1,526}, (2,2,317}, (3,0,63834}, (3,3,63761}, (4,4,55552}

  The appearance of (n,m,k) in the next list means that X_k = tan(nB+mC) + tan(mB+nC):

  (1,0,3), (1,1,4), (2,0,1147), (2,1,1154), (2,2,68), (3,0,63837), (3,3,562), (4,4,43973)

  For midpoints of trigonometric points, see the preamble just before X₆₃₈₃₉.

• X₆₃₈₃₉–X₆₃₈₄₆ — Trigonometric Midpoints

  contributed by Clark Kimbering and Peter Moses, June 17, 2024. The appearance of (n,m,k) in the following list means that X_k = midpoint of sin(nB+mC) : : and sin(mB+nC) : : .

  (1,0,10), (1,2,25639), (1,-2,3814), (1,-3,34825), (2,0,5), (2,-4,34826), (3,0,63803), (3,1,63840), (3,-2,63841), (4,0,5449), (4,2,63839)

  The appearance of (n,m,k) in the following list means that X_k = midpoint of cos(nB+mC) : : and cos(mB+nC) : : .

  (1,0,142), (1,1,226), (2,0,13567), (2,1,63844), (2,2,343), (3,1,63843), (4,0,63842), (4,4,63806)

  The appearance of (n,m,k) in the following list means that X_k = midpoint of cot(nB+mC) : : and cot(mB+nC) : : .

  (1,1,6), (1,2,3284), (1,-2,216), (2,2,577), (3,1,63845), (3,3,63834), (3,-1,63846)

  Midpoints of pairs such as tan(kB)+tan(kC) : : and tan(kC)+tan(kB) : : can be found elsewhere in ETC using the identity tan(kB)+tan(kC) = sin(2kA); e.g., X₆₃₈₃₇ = sin(6A) : : .

  For more trigonometric triangle centers, see the preamble just before X₆₃₈₂₅.

• X₆₃₉₆₁–X₆₄₀₀₀ — Points related to crosspedal triangles, Part 1

  contributed by Ivan Pavlov on June 29, 2024. [Editor’s note: soon after acceptance, the centers somehow got lost. After recovery, they were published here on January 25, 2025.] Given a triangle ABC and two points P and Q not on its sides, let the line through Q parallel to AP intersect lines AB and AC at points A_b and A_c. Similarly define B_a, B_c, C_a, C_b. The lines B_aC_a, A_bC_b, A_cB_c form a triangle called here the P-crosspedal triangle of Q. We remind the reader of two other definitions used for these centers:

  1. Through Q construct a line parallel to AP and let A' be the intersection point with BC. Similarly define B' and C'; A'B'C' is called P-pedal triangle of Q.

  2. The P-antipedal triangle of Q is the triangle A'B'C' such that ABC is P-pedal of Q wrt A'B'C'.

  For more information and properties see Euclid 6286

Part 33

• X₆₄₄₂₆–X₆₄₄₃₇ — Composites

  contributed by Clark Kimberling and Peter Moses, July 3, 2024. Suppose that P =\(p(a,b,c) : p(b,c,a) : p(c,a,b)\) and U =\(u(a,b,c) : u(b,c,a) : u(c,a,b)\) are triangle centers, where p(a,b,c) and u(a,b,c) are polynomials in standard form (i.e., p(a,b,c) and p(b,c,a) are relatively prime, and the coefficient of the highest power of a is positive, or if p(a,b,c) is invariant of a then the coefficient of highest power of b is positive.)

  Define the composite P-of-U to be the triangle center given by

  P-of-U = p(u(a,b,c), u(b,c,a), u(c,a,b)) : p(u(b,c,a), u(c,a,b), u(a,b,c)) : p(u(c,a,b), u(a,b,c), u(b,c,a)).

  For example, X₃-of-X₃ = X₁₁₄₇ = a^4(a^2 - b^2 - c^2)(a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2) : :

  Suppose next that the Euler line is represented as a linear combination of X₃ and X₄ as follows:

  V(r,s) = a^2 (a^2 - b^2 - c^2)*r + (b^2 - c^2 - a^2)(c^2 - a^2 - b^2)*s : : ,

  where r and s are not both 0. Let P = X₁₀ = b + c : c + a : a + b.

  Then “X₁₀-of-Euler-line” is the line given by

  X₁₀-of-V(r,s), which as the linear combination

  (a^2 (b^2 + c^2) + (b^2 - c^2)^2)*r - 2 a^2(a^2 - b^2 - c^2)*s : :

  is essentially X₅*r + X₃*s, the Euler line.

  Reversing the order of the composition gives “the Euler line of X₁₀”, consisting of points

  W(r,s) = (b + c)^2 (a^2 - b c + a b + a c)*r + 2(b^2 - c a + b c + b a)(c^2 - a b + c a + c b)*s : :,

  which is essentially X₄₀₇₅*r + X₅₉₆*s.

  The appearance of (r,s,k) in the following list means that r and s are not both 0 and W(r,s) = X_k:

  (r,0,4075), (0,s,596), (r,r,6532), (r,-r,6534), (2r,r,2), (2r,-r,24068), and in general, we have the combo

  W(r,s) = 3*r*X₂ + (3*s - r)*X₅₉₆. The list continues:

  (-4,1,46426), (-10,3,46427), (-3,1, 46428), (-2,3,46429), (1,2,46430), (2,3,46431), (4,3,46432), (3,2,46433), (3,1,46434), (10,3,46435), (-1,1,46436), (-1,3,46437).

  A selection of points of the form X_i-of-X_i appear next: X₁-of-X₁ = X₁

  X₂-of-X₂ = X₂

  X₃-of-X₃ = X₁₁₄₇

  X₄-of-X₄ = X₃₃₄₆

  X₅-of-X₅ = X₆₄₄₅₂

  X₆-of-X₆ = X₃₂

  X₇-of-X₇ = X₁₀₄₀₅

  X₈-of-X₈ = X₁₄₅

  X₉-of-X₉ = X₁

  X₁₀-of-X₁₀ = X₁₁₂₅

  [and others].

  The appearance of (i,j,k) in the following list means that X_i-of-X_j = X_k:

  (3,3,1147), (3,4,6523), (3,5,6663), (3,6,206), (3,7,17113), (3,8,6552), (3,9,6600), (3,10,4075), (3,11,64440), (4,1,4), (4,2,2), (4,3,68), [and others].

• X₆₄₄₄₆–X₆₄₄₅₁ — Miyamoto-mixtilinear related centers

  contributed by César Eliud Lozada, July 8, 2024. The following conjectures are due to Keita Miyamoto, July 02, 2024: In a triangle ABC with circumcircle ω, denote:

  .LI li{padding-bottom:6px; before.content: counter(list, lower-alpha) “) “;} \(a_{1}\), \(b_{1}\), \(c_{1}\): the A-, B-, C- mixtilinear excircles of ABC, respectively. \(o_{1}\): the outer-Apollonius circle of \(a_{1}\), \(b_{1}\), \(c_{1}\). \(a_{2}\): the circle, other than ω, passing through B and C and internally tangent to \(a_{1}\). Cyclically \(b_{2}\) and \(c_{2}\). \(o_{2}\): the outer-Apollonius circle of \(a_{2}\), \(b_{2}\), \(c_{2}\). A_b, A_c: the second intersections of \(a_{2}\) and AC, AB, respectively. Similarly B_c, B_a and C_a, C_b. A': the second intersection of \(b_{2}\) and \(c_{2}\), and, cyclically B', C'. \(t_{a}\): the common tangent of \(a_{1}\) and \(a_{2}\), and, cyclically \(t_{b}\), \(t_{c}\).

  Then: (a) \(o_{1}\) and \(o_{2}\) are tangent. (b) A'B'C' and the triangle bounded by the lines A_bA_c, B_cB_a, C_aC_b are perspective. (c) ABC and the triangle bounded by the lines \(t_{a}\), \(t_{b}\), \(t_{c}\) are perspective.

  Results:

  The center of the circle \(o_{1}\), found by Peter Moses, is X₈₁₅₈. The touchpoints of the mixtilinear excircles and \(o_{1}\) are the vertices of the 9th mixtilinear triangle. See X₈₁₅₈. The center of the circle \(o_{2}\), found by Angel Montesdeoca, is X₃₅₅₉₉. See X₃₅₅₉₉. The touchpoint of \(o_{1}\), \(o_{2}\), in (a), is X₆₄₄₄₆. The perspectors in (b) and (c) are X₆₄₄₄₇ and X₆₄₄₄₈, respectively.

  Similar points can be found by using mixtilinear incircles instead of mixtilinear excircles and inner-Apollonius circles instead of outer Apollonius circles. With this new construction: The center of the circle \(o_{1}\), found by Peter Moses, is X₆₇₆₇. The touchpoints of the mixtilinear incircles and \(o_{1}\) are the vertices of the 8th mixtilinear triangle. See X₆₇₆₇. The center of the circle \(o_{2}\) is X₆₄₄₄₉. The touchpoint of \(o_{1}\), \(o_{2}\), in (a), is X₄₄₈₅₈. The perspectors in (b) and (c) are X₆₄₄₅₀ and X₆₄₄₅₁, respectively.

• X₆₄₅₂₂–X₆₄₅₈₄ — Points related to the 1st Pavlov-Altintaş triangle

  contributed by Ivan Pavlov on July 25, 2024. Let I_aI_bI_c be the intouch triangle. Let AI intersect I_aI_b and I_aI_c at points A_b and A_c resp. Let N_a be the nine-point center of I_aA_bA_c and similarly define N_b and N_c. In the following, N_aN_bN_c is called the 1st Pavlov-Altintaş triangle. Its inverse triangle is called the 1st anti-Pavlov-Altintaş triangle. The barycentric coordinates of their A-vertices are

  1st Pavlov-Altinta ş : -a*(b-c)^2 : b*((b-c)*c+a*(b+c)) : c*(b*(-b+c)+a*(b+c))

  1st anti-Pavlov-Altinta ş : a-b-\(c : c : b\)
  The 1st Pavlov-Altinta ş triangle is perspective to ABC with perspector X₁₃₄₇₆. It is also bilogic to the intouch triangle and has the same centroid - X₃₅₄. The perspector of the intouch and 1st Pavlov-Altinta ş triangles is an infinite point, X₅₁₃. Some other triangles perspective to the 1st Pavlov-Altintas include: AAOA, Aquila, Artzt, 8th Brocard, circumsymmedial, Lucas central, Lucas inner, Lucas reflection, Lucas tangents, 1st Pamfilos-Zhou, 1st and 2nd Sharygin, symmedial, tangential, inner-Yff, outer-Yff.

  The 1st anti-Pavlov-Altinta ş triangle is perspective to ABC with perspector X₇₅. Some other triangles perspective to it include: Gemini 16, Gemini 17, Gemini 111, Aquila, inner-Conway, inner-Garcia, Yff contact, inner-Yff, outer-Yff.

  The 1st anti-Pavlov-Altinta ş triangle is orthologic to the intouch with orhtology center X₃₈₆₉. It is also orthologic to the 5th Conway and 1st Savin triangles with orthology centers resp. X₆₄₀₀₂ and X₈.

  For more information on some related triangles see this Euclid thread.

• X₆₄₆₉₃–X₆₄₇₆₈ — Points related to crosspedal triangles, Part 2

  contributed by Ivan Pavlov on August 06, 2024. Given a triangle ABC and two points P and Q not on its sides, let the line through Q parallel to AP intersect lines AB and AC at points A_b and A_c. Similarly define B_a, B_c, C_a, C_b. The lines B_aC_a, A_bC_b, A_cB_c form a triangle called here the P-crosspedal triangle of Q.

  We remind the reader of two other definitions used for these centers:

  1. Through Q construct a line parallel to AP and let A' be the intersection point with BC. Similarly define B' and C'; A'B'C' is called P-pedal triangle of Q

  2. The P-antipedal triangle of Q is the triangle A'B'C' such that ABC is P-pedal of Q wrt A'B'C'.

  For more information and properties see Euclid 6286

• X₆₄₈₅₅–X₆₄₈₈₉ — Points on the line at infinity

  contributed by Clark Kimberling and Peter Moses, August 21, 2024. Suppose that X =\(x:y:z\) is a point on the infinity line. Then the following points are also on the infinity line:

  y sin B - z sin C : : y tan B - z tan C :: y sec B - z sec C : :

  The appearance of (i,j) in the following list means that if X_i =\(x:y:z\), then X_j = y sin B - z sin C : :

  (511,64855), (512,714), (513,726), (514,536), (516,64856), (517,64857), (518,522), (519,4777), (521,64858), (522,518), (523,740), (524,64859), (525,8680), [and others].

  The appearance of (i,j) in the following list means that if X_i =\(x:y:z\), then X_j = y tan B - z tan C : :

  (30,9007), (511,520), (512,8681), (513,34381), (514,9028), (517,9051), (518,521), (519,9031), (520,511), (521,518), (522,64875), (523,3564), (524,525), (525,524), [and others].

  The appearance of (i,j) in the following list means that if X_i =\(x:y:z\), then X_j = y sec B - z sec C : :

  (513,9028), (514,912), (515,64885), (518,64886), (520,8680), (521,527), (522,64887), (523,64888), (525,758), (527,521), (758,525), (812,64889), (912,514)

• X₆₄₉₇₀–X₆₅₀₈₄ — Basepoints of perspective triangles

  contributed by Ivan Pavlov on Aug 26, 2024. Given two perspective, but non-homothetic central triangles P₁P₂P₃ and Q₁Q₂Q₃ with perpsector S, determine the numbers \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that:

  \(x_{i}\)*S + (1-\(x_{i})\)*Q_i = P_i for each i=1,2,3 where the sum and equality are barycentric operations.

  If such numbers exist they are unique and (\(x_{1}:x_{2}:x_{3}\)) is a triangle center which we call 1st basepoint of P₁P₂P₃ wrt Q₁Q₂Q₃.

  Similarly, using the conditions (1-\(x_{i})\)*S + \(x_{i}\)*Q_i = P_i, we can define the 2nd basepoint.

  It can be proven that if the 1st basepoint exists, then the 2nd basepoint also exists.

  For more information and results see this Euclid thread.

• X₆₅₀₉₆–X₆₅₁₀₆ — Sign-images

  contributed by Clark Kimberling and Peter Moses, August 28, 2024. Suppose that X = x(a,b,c) : : is a triangle center, and define

  \(f(a,b,c) = x(a\),-b,c) and \(g(a,b,c) = x(a\),b,-c)

  X'(a,b,c) =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\)

  X″(a,b,c) =\(g(a,b,c) : g(b,c,a) : g(c,a,b)\).

  The point X' = X'(a,b,c) is here introduced as the sign-image of X. The set of triangle centers is partitioned by the sign-image operation into three subsets:

  Type 1: self-sign-images X, for which X'=X;

  Type 2: triangle centers X such that X' ≠ X and X' = X″;

  Type 3: triangle centers X such that X' ≠ X″. In this case X' and X″ are a bicentric pair.
  The appearance of k in the following list means that X_k is self-sign-image:

  1, 2, 3, 4, 5, 6, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 30, 31, 32, 38, 39, 47, 48, 49, 50, 51, 52, 53, 54, 61, 62, 63, 64, 66, 67, 68, 69, 70, 74, 75, [and others].
  The appearance of (h,k) in the following list means that X_k is the sign-image of X(h) and X_k is a triangle center:

  (7,8), (10,514), (12,11), (34,33), (35,36), (36,35), (37,513), (42,649), (56,55), (57,9), (65,650), (71,1459), (72,905), (73,652), (77,78), (84,40), (85,312), (87,43), (90,46)
  The appearance of k in the following list means that X'(k) and X″(k) are a bicentric pair:

  8, 9, 11, 21, 27, 28, 29, 33, 40, 41, 43, 44, 45, 46, 55, 58, 59, 60, 78, 79, 80, 81, 86, 88, 89, 100, 101, 102, 103, 104, 105, 106, 108, 109, 116, 117, 118, 119, 120, [and others].
  Regarding triangfle centers of Type 2, the appearance of {j,k} in the following list means that X_k is the sign-image of X_j and X_j is the sign-image of X_k:

  {10,514}, {35,36}, {37,513}, {42,649}, {71,1459}, {72,905}, {171,238}, {172,1914}, {202,7006}, {203,7005}, {213,667}, {228,22383}, {239,894}, [and others]

• X₆₅₁₁₂–X₆₅₁₁₈ — Double-sign-images

  contributed by Clark Kimberling and Peter Moses, August 31, 2024. Suppose that X = x(a,b,c) : : is a triangle center, and define

  \(f(a,b,c) = x(a\),-b,-c)

  X*(a,b,c) =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\)

  The point X* = X*(a,b,c) is here introduced as the double-sign-image of X. The set of triangle centers is partitioned by the double-sign-image operation into two subsets:

  (1) double-self-sign-images X, for which X*=X;

  (2) triangle centers X such that X* ≠ X.

  The appearance of (h,k) in the following list means that X_k = X*(h): (8,2), (9,1), (11,1086), (21,100), (27,1897), (28,1783), (29,1897), (33,19), (40,1), (41,31), (43,1), (44,1100), (45,16777), (46,1), (55,6), (58,101), (59,7341), (60,1252), (78,63), (79,80), (80,79), (81,100), (86,190), (88,65112), (89,65113), (100,81), (101,58), (104,65115), (108,1396), (109,1412), (116,65116), (121,65117), (124,65118)

• X₆₅₃₈₆–X₆₅₅₂₄ — Pairs of triangles with a common inconic

  contributed by César Eliud Lozada, September 24, 2024. The following proposition is proved in Macaulay, F.S., Geometrical conics, Cambridge, 1895, pp 241:

  Prop. 77: If two triangles are circumscribed to a conic, they are also inscribed to a conic; and conversely.

  In the preamble just before X₁₄₇₁₃, there were described a set of conics circumscribing some selected pairs of triangles. Then, according to the reciprocal of the previous proposition, all pairs of triangles having a common circumconic also have a common inconic.

  Let T=ABC, T'=A'B'C' be two triangles. The cross-triangle of T and T' is defined as the triangle A*B*C*, with A*=BC'∩B'C, B*=CA'∩C'A and C*=AB'∩A'B. Then, by Brianchon theorem, the points A, B, C, A″, B″, C″ lie on a conic if A*, B*, C* are collinear or coincident points, i.e., if A*B*C* is a degenerate triangle. This is the required condition for T and T' to be inscribed in a common conic and, consequently, to be circumscribed to another common conic.

  An extensive list of centers of inconics of pairs of triangles can be seen here. For definitions of all triangles listed here, check the Index of triangles referenced in ETC.

• X₆₅₅₂₅–X₆₅₅₆₂ — Centers of common circumconics of two triangles (2)

  contributed by César Eliud Lozada, September 27, 2024. This section continues the section

• X₁₄₇₁₃–X₁₄₇₈₁ — Centers of common circumconics: X₁₄₇₁₃-X₁₄₇₈₁

  . For definitions of all triangles in this section, check the Index of triangles referenced in ETC.

• X₆₅₃₃₀–X₆₅₅₃₃ — In-Exsimilicenters of Hatzipolakis-Suppa Circle and some circles

  contributed by Ercole Suppa, October 8, 2024.

  X(m,n) = (insimilicenter, exsimilicenter) of circles

  Hatzipolakis-Suppa circle, anticomplementary circle X(3543, 3146)

  Hatzipolakis-Suppa circle, circumcircle X (4, 30)

  Hatzipolakis-Suppa circle, 2nd Lemoine circle (or cosine circle) X(35820, 35821)

  Hatzipolakis-Suppa circle, half-Moses circle X(65630, 44526)

  Hatzipolakis-Suppa circle, incircle X(12943,12953)

  Hatzipolakis-Suppa circle, Johnson triangle circumcircle X(3627, 30)

  Hatzipolakis-Suppa circle, 1st Johnson-Yff circle X(65631 ,6284)

  Hatzipolakis-Suppa circle, 2nd Johnson-Yff circle X(7354,65532)

  Hatzipolakis-Suppa circle, Moses circle X(62203, 65633)

  Hatzipolakis-Suppa circle, nine-point circle X(4, 20)

  Hatzipolakis-Suppa circle, sine triple angle circle X(6759, 13352)

  Hatzipolakis-Suppa circle, Stammler circle X(3830, 5073)

  Hatzipolakis-Suppa circle, Steiner circle X(17578, 33703)

  Hatzipolakis-Suppa circle, incircle of orthic triangle X(44438, 12173)

  Hatzipolakis-Suppa circle, Lucas inner circle X(8981, 9541)

  Hatzipolakis-Suppa circle, Lucas radical circle X(6564, 42266)

  Hatzipolakis-Suppa circle, Lucas(-1) inner circle X(?, 13966)

  Hatzipolakis-Suppa circle, Lucas(-1) circles radical circle X(42267, 6565)

  For definitions of circles listed here, check the Extended glossary.

• X₆₅₆₃₅–X₆₅₆₅₀ — P-antipodes of points on the circumcircle

  contributed by Clark Kimberling and Peter Moses, October, 2024. Suppose that P is a point on the circumcircle of a triangle ABC, and that U is the isogonal conjugate of P, so that U is on the line at infinity. The U-antipode of P is the point, other than P, in which the line PU meets the circumcircle. If P =\(p : q : r\) (on the circumcircle), then a 1st barycentric for the U-antipode of P is given by

  a^2*q*r/(a^2*q*(q - r)*r - p*(q + r)*(c^2*q - b^2*r)) : : .

  this being the Collings transform of the isogonal conjugate of P.

  The appearance of (i,j,k) in the following list means that X_k = X_j-antipode of X_i and k < 65000: (74,30,476), (98,511,805), (99,512,805), (100,513,901), (101,514,927), (102,515,1309), (103,516,927), (104,517,901), (105,518,6078), (106,519,6079), (107,520,6080), (108,521,6081), (109,522,1309), (110,523,476), (111,524,6082), (112,525,2867), (476,526,16170), (477,5663,16170), [and others].

  The appearance of (i,j,k) in the following list means that X_k = X_j-antipode of X_i and k > 65000: (741,740,65635), (813,812,65636), (840,528,65637), (843,543,65638), (901,900,65639), [and others].

• X₆₅₆₅₁–X₆₅₇₀₉ — Crosshexagon points and lines

  contributed by César Eliud Lozada, October 13, 2024. Let T' = A'B'C' and T″ = A″B″C″ be two triangles circumscribed by a conic. Let A_b = B'C'∩A″C″ and A_c = B'C'∩A″B″, and denote B_c, C_a and B_a, C_b cyclically. Let P_a = A_bB_c∩A_cC_b, and define P_b, P_c cyclically. Then: Lines B_cC_b, C_aA_c, A_bB_a concur in a point 𝒳. Points P_a, P_b, P_c lie on a line ℒ.

  The hexagon with vertices {A_b, A_c, B_c, B_a, C_a, C_b} is named here the crosshexagon of T' and T″, and 𝒳 and ℒ will be referred here as the crosshexagon point and crosshexagon line, respectively, of T' and T″.

  Lists of calculated and not calculated crosshexagon points are available here (X(65651).txt). For definitions of all triangles referred here, check the Index of triangles referenced in ETC.

• X₆₅₇₄₂–X₆₅₇₅₂ — D-maps

  contributed by Clark Kimberling and Peter Moses, October 17, 2024. Suppose that U and V are given by normalized barycentrics U = (u,v,w) and U' = (u’,v’,w’). The D-map of U and V is here introduced as the point \(u'' : v'' : w''\) given by

  u’’ = \(S_{A}(u - u')\)², v’’ = \(S_{B}(v - v')\)², w’’ = \(S_{C}(w - w')\)².

  The name D-map corresponds to the fact that |UU'|² = u’’+v’’+w’’. If U and U' lie on a line L, then every pair of points on L have the same D-map. If U and U' are triangle centers, then the D-map of U and U' are triangle centers.

  The appearance of (i,j,k) in the following list means that the D-map of X_i and X_j is X_k, where k < 60000.

  (1, 4, 38554)

  (2, 3, 16163) (Euler line)

  (3, 10, 38554)

  (8, 20, 38554)

  (13, 15, 16163)

  (14, 16, 16163)

  (44, 513, 3937) (anti-orthic axis)

  (230, 231, 125) (orthic axis)

  (241, 514, 1565) (Gergonne line)

  (325, 523, 125) (de Longchamps axis)

  (522, 650, 2968) (Garcia-Reznik line)

  (513, 663, 3937) (Helman line)

  The appearance of (i,j,k) in the following list means that the D-map of X_i and X_j is X_k, where k > 60000.

  (1, 2, 65742)

  (1, 3, 65743)

  [and others].

• X₆₅₇₈₄–X₆₅₈₅₂ — Bicevian-, bianticevian-, bipedal- and biantipedal- inconics

  contributed by César Eliud Lozada, October 19, 2024. Let ABC be a triangle and P', P″ two distinct points. The following facts are widely known: The circumcevian triangles T', T″ of P', P″, respectively, are inscribed in the circumcircle of ABC. The circumanticevian triangles T', T″ of P', P″, respectively, are inscribed in the circumcircle of ABC. The cevian triangles T', T″ of P', P″, respectively, are circumscribed by a conic, named the bicevian conic of P’ and P”. The anticevian triangles T', T″ of P', P″, respectively, are circumscribed by a conic, named the bianticevian conic of P’ and P”. If P', P″ are isogonal conjugates, the pedal triangles T', T″ of P', P″, respectively, are circumscribed by a circle, named the pedal circle of P’ or P”. If P', P″ are isogonal conjugates, the antipedal triangles T', T″ of P', P″, respectively, are circumscribed by a conic, named the antipedal conic of P’ or P”, or the apedal conic of P’ or P” (see preamble before X₈₂₆₈).

  In all the previous cases, there are two triangles inscribed in a conic. Therefore, by the proposition if two triangles are circumscribed to a conic, they are also inscribed to a conic; and conversely, used in the preamble before X₆₅₃₈₆, every pair of those of triangles are tangent to a common conic, here accordingly named, the bicircumcevian-, bicircumanticevian-, bicevian-, bianticevian-, bipedal- and biantipedal- inconic of T' and T″.

  This section includes the centers and perspectors of these inconics for P'=X_i and P″=X_j, with 1≤{i, j}≤11. A list of already known centers and perspectors can be seen here (X(65784).pdf).

  As a remarkable note, for any P' and P″, ABC is autopolar with respect to the bicevian inconic of P' and P″.

• X₆₅₈₈₇–X₆₅₉₄₇ — W-maps

  contributed by Clark Kimberling and Peter Moses, October 22, 2024. Suppose that distinct points U and U' are given by normalized barycentrics U = (u,v,w) and U' = (u’,v’,w’). The W-map of U and U' is here introduced as the point W(U,U') given by

  W = W(U,U') = (u - u’)² : (v - v’)² : (w - w’)².

  As a barycentric square, the point W lies on the Steiner inellipse, and the point W* = u-u’ : v-v’ : w-w’ lies on the line at infinity. Let g(W*) = isogonal conjugate of W*, so that g(W*) lies on the circumcircle. Then W = crosspoint(X₂ and W*) = crosssum(X₆ and g(W*)).

  The W-map is related to the D-map introduced in the preamble just before X₆₅₇₄₂. If U and U' lie on a line L, then every pair of points on L have the same W-map. Thus, W(U,U') can be regarded as a mapping from the line L = UU' to W(U,U'), for which we write W(L), as well as W(U,U'). If U and U' are triangle centers, then W(U,U') is a triangle center.

  W(Euler line) = W(X₂,X₃) = X₃₁₆₃ = crosssum of X₆ and X₇₄

  W(Brocard axis) = W(X₃,X₆) = X₁₁₆₇₂ = crosssum of X₆ and X₉₈

  W(IO line) = W(X₁,X₃) = X₂₃₉₈₀ = crosssum of X₆ and X₁₀₄

  W(orthic axis) = W(X₂₃₀,X₂₃₁) = X₁₁₅ = crosssum of X₆ and X₁₁₀

  W(anti-orthic axis) = W(X₄₄,X₅₁₃) = X₁₀₁₅ = crosssum of X₆ and X₁₀₀

  W(Lemoine axis) = W(X₁₈₇,X₂₃₇) = X₁₀₈₄ = crosssum of X₆ and X₉₉

  W(de Longchamps axis) = W(X₃₂₅,X₅₂₃) = X₁₁₅ = crosssum of X₆ and X₁₁₀

  W(Gergonne line) = W(X₂₄₁,X₅₁₄) = X₁₀₈₆ = crosssum of X₆ and X₁₀₁

  W(Soddy line) = W(X₁,X₇) = X₂₃₉₇₂ = crosssum of X₆ and X₁₀₃

  W(Nagel line) = W(X₁,X₂) = X₄₃₇₀ = crosssum of X₆ and X₁₀₆

  W(Fermat line) = W(X₆,X₁₃) = X₂₃₉₆₇ = crosssum of X₆ and X₈₄₂

  W(Napoleon axis) = W(X₆,X₁₇) = X₆₅₉₁₇ = crosssum of X₆ and X₅₉₆₆

  W(van Aubel line) = W(X₄,X₆) = X₂₃₉₇₆ = crosssum of X₆ and X₁₂₉₇

  [and others].

  The appearance of (i,j,k) in the following list means that W(X_i,X_j) = X_k, where k < 65887. (1,2,4370), (1,3,23980), (1,4,23986), (1,5,61066), (1,6,6184), (1,7,23972), (1,21,35069), (1,79,3163), (1,75,35068), (1,87,20532), (1,88,35129), (1,142,35111), (1,147,35082), (1,190,35123), (2,3,3163), (2,6,2482), (2,7,35110), (2,11,35113), (2,13,61068), (2,14,61069), [and others].

  The appearance of (i,j,k) in the following list means that W(X_i,X_j) = X_k, where k > 65886.

  (1,19,65887), (1,39,65888), (1,41,65889), (1,76,65890), (1,84,65891), (1,85,65892), (1,90,65893), (1,99,65894), (1,104,65895), (2,12,65896), (2,17,65897), (2,18, 65898), [and others].

• X₆₅₉₈₅–X₆₆₀₇₁ — Points releated to the orthic-of-Fuhrmann triangle

  contributed by Ivan Pavlov on October 30, 2024. For more information and constructions of the orthic-of-Fuhrmann triangle see this Euclid thread.

Part 34

• X₆₆₀₈₇–X₆₆₀₈₉ — Points releated to the orthic-of-Fuhrmann triangle,Points with coordinates generated using a Perspective Fieldc

  contributed by Chris van Tienhoven, December 8, 2024. Points X₆₆₀₈₇-X₆₆₀₈₉ can be calculated in a simple form using the concepts of Perspective Fields. For more information, see the introduction of ETC (November 22, 2024) and the links provded there.

  Advantages of Perspective Fields

  One advantage of Perspective Fields is that, once the perspective coordinates (\(n1: n2: n3\)) of a point are known in a Perspective Field [P₁, P₂, P₃; P₄], the actual coordinates can be calculated using this formula:

  Px = n1.det[P₄, P₂, P₃].P₁ + n2.det[P₁, P₄, P₃].P₂ + n3.det[P₁, P₂, P₄].P₃.

  This is particularly useful when working with points whose coordinates are given by elaborate trigonometric functions. By associating such points within a Perspective Field (PF) defined by four reference points—typically with simple coordinates—the calculations yield relatively straightforward results, as indicated by these three examples:

  X₆₆₀₈₇ has perspective coordinates (\(1 : 2 : 0\)) in PF[X₃, X₃₅₆, X₁₁₃₄; X₁₁₃₇].

  X₆₆₀₈₉ has perspective coordinates (\(1 : -1: 1\)) in PF[X₄, X₃₅₆, X₁₁₃₄; X₃₂₇₉].

  X₅₃₉₀ has perspective coordinates (\(2 : 1 : 2\)) in PF[X₂, X₁₁₃₆, X₁₁₃₇; X₃₂₇₃].

  Finding a Perspective Field for a point

  To determine the Perspective Field for a point, its perspective coordinates are calculated in different numeric configurations of two reference triangles. If the coordinates match for both configurations, the point is part of the Perspective Field, and its perspective coordinates are already known from previous calculations. Although it is impractical to perform theses calculations for many sets of four points in ETC, there are certainly many other sets of four points that are amenable to Mathematica, Maple, etc.

• X₆₆₀₉₂–X₆₆₁₀₉ — Points on Terzić hyperbolas

  contributed by Clark Kimberling and Peter Moses, November 7, 2024. Early in November, 2024, Predrag Terzić contributed notes on three hyperbolas, and Peter Moses found equations and pass-through points for the hyperbolas.

  Points X₆₆₀₉₂-X₆₆₀₉₇, along with the points X_i for i = 5, 13, 14, 15, 16, 549, lie on the 1st Terzić hyperbola, given by the following barycentric equation: [long equation omitted here].

  The center of this hyperbola is X₃₀₅₅.

  Points X₆₆₀₉₈-X₆₆₁₀₂, along with the points X_i for i = 1, 4, 9, 13, 14, 321, lie on the 2nd Terzić hyperbola, given by the following barycentric equation: [long equation omitted here].

  Points X₆₆₁₀₃-X₆₆₁₀₉, along with the points X_i for i = 3, 4, 10, 13, 14, 386, lie on the 3rd Terzić hyperbola, given by the following barycentric equation: [long equation omitted here].

• X₆₆₁₁₀–X₆₆₁₁₄ — Points on the Lester circle

  contributed by Clark Kimberling and Peter Moses, November 7, 2024. Indices i < 40000 such that X_i lies on the Lester circle: 3, 5, 13, 14, 1117, 5671, 14854, 15475, 15535, 15536, 15537, 15538, 15539, 15540, 15541, 15542, 15543, 15544, 15545, 15546, 15547, 15548, 15549, 15550, 15551, 15552, 15553, 15554, 15555, 34365

  Indices i > 40000 such that X_i lies on the Lester circle: 66110, 66111, 66112, 66113, 66114

• X₆₆₁₉₃–X₆₆₂₅₉ — Points releated to the 1st Van-Khea-Pavlov triangle

  contributed by Ivan Pavlov on Nov 13, 2024. Let P_aP_bP_c be the intouch triangle. Let A_b and A_c be the reflections of P_a in the midpoints of BP_b and CP_c. Let A_bA_c intersect P_bP_c at point T_a, and similarly define T_b and T_c. T_aT_bT_c is homothetic to the excenters-midpoints triangle with center X₅₅ and ratio r/R. It is bilogic to the following triangles: ABC, Garcia-reflection, 1st Pavlov, orthic-of-Fuhrmann. We call T_aT_bT_c the 1st Van-Khea-Pavlov triangle. For more information see this Euclid thread.

• X₆₆₄₂₆–X₆₆₄₇₄ — Points releated to the 2nd outer-Grebe triangle

  contributed by Ivan Pavlov on Nov 25, 2024. On the sides of ABC, construct squares ABCbCa, BCAcAb, and CABaBc. The triangle formed by lines B_aC_a, A_bC_b, and A_cB_c is called here the 2nd outer-Grebe triangle.

  It is homothetic to the Artzt triangle and the center of homothety is X₆₈₁₁. For more information about the 2nd outer-Grebe triangle see this Euclid thread.

• X₆₆₅₃₂–X₆₆₅₄₇ — Points related to the Kirikami-Steiner trifolium

  contributed by Ivan Pavlov on Dec 18, 2024. Let P be a point not on the sides of ABC and G its centroid. Denote A' = Kirikami center of PBCQ, A″ = Kirikami center of PCBQ and similarly define B',B″,C',C″. The locus of points P for which A'B'C' and A″B″C″ are perspective is a circumquartic, which is called here the Kirikami-Steiner trifolium. It is tangent to the Steiner circumellipse at A, B, and C and has a triple point at G. The Kirikami-Steiner trifolium is the inverse image of K015 in the Steiner circumellipse. In the following list (i,j) means that for P=X_i the perspector of A'B'C' and A″B″C″ is X_j: (1,66543), (3,66544), (6,66545), (37,66546), (39,66547). For more information see this Euclid post.

• X₆₆₅₅₄–X₆₆₅₈₀ — Perspectors related to Morley triangles

  :contributed by César Eliud Lozada, December 23, 2024. (Many thanks to Chris van Tienhoven for sharing his deep knowledge on this topic.) Two triangles related to the original Morley triangles are recovered and renamed here: The Morley-homothetic-inscribed triangle is the triangle inscribed in ABC and homothetic to the 1st-, 2nd- and 3rd- Morley triangles. This triangle was introduced by Peter Moses as triangle J₁J₂J₃ just before X₃₂₇₂. Its center is precisely X₃₂₇₂ and it has A-vertex with trilinear coordinates: J₁ = 0 : sin((A - C + π)/3) : sin((A - B + π)/3) (See Peter Moses, X₃₂₇₂) The Morley-homothetic-circumscribed triangle is the triangle circumscribed to ABC and homothetic to the 1st-, 2nd- and 3rd- Morley triangles. This triangle is cited by Randy Hutson in a property of X₈₀₁₁. Its center is X₈₀₁₁ and it has A-vertex with trilinear coordinates: A″ = -1 : (a*(-1+4*z^2)-(2*(y-2*x*z))*c)/((-1+4*z^2)*b-(2*(x-2*y*z))*c): (a*(-1+4*y^2)-(2*(z-2*x*y))*b)/((-1+4*y^2)*c-(2*(x-2*y*z))*b)

  where x = cos(A/3), y = cos(B/3), z = cos(C/3)

  or, equivalently, A″ = cos((2*(B-C))/3)+2*cos((B-C)/3)*sin(A+π/6) :

  -2*csc((B-C+π)/3)*sin((A-C+π)/3)*(sin(C)*sin((B-A+π)/3)+sin(A)*sin((B-C+π)/3)) :

  -2*csc((C-B+π)/3)*sin((A-B+π)/3)*(sin(B)*sin((C-A+π)/3)+sin(A)*sin((C-B+π)/3))

• X₆₆₆₃₀–X₆₆₈₆₉ — Points related to some P-anticomplementary triangles

  contributed by Ivan Pavlov on Jan 14, 2025. Let ABC be a triangle and P a point and P_aP_bP_c the cevian triangle of P. Denote by M_a, M_b, M_c the midpoints of AP_a, BP_b, CP_c, resp. and by P₁, P₂, P₃ the the isogonal conjugates of P wrt triangles M_aBC, M_bCA, M_cAB, resp. If P lies on the circumcircle, P₁P₂P₃ is similar to ABC and its circumcircle is the anticomplementary circle.

  This fact suggests that for each point P we name the triangle P₁P₂P₃ - P-anticomplementary triangle. Some other properties, if P lies on the circumcircle, include:

  - The orthology center of ABC and P₁P₂P₃ is Λ(X₃, P), and lies on the circumcircle.

  - The orthology center P₁P₂P₃ and ABC is the reflection of X₂₀ in P and lies on the anticomplementary circle.

  - The X₁-anticomplementary triangle is homothetic to the Artzt triangle with center X₆₆₆₃₂

  - The X₄-anticomplementary triangle is homothetic to the orthic triangle with center X₂₀

  Some of the properties below refer to CTR-triangles. More info on these series of triangles is available in this catalog.

• X₆₆₉₇₅–X₆₆₉₈₉ — Points on the Moses Inparabola

  contributed by Clark Kimberling, January 25, 2025. This preamble is based on notes received from Peter Moses, January 23-25, 2025. In the plane of a triangle ABC, the Moses inparabola is here introduced as the curve given by the following barycentric equation:

  (a - b - c)^2*(b - c)^2*x^2 - 2*(a - c)*(a - b - c)*(b - c)*(a - b + c)*x*y + (a - c)^2*(a - b + c)^2*y^2 + 2*(a - b)*(a - b - c)*(b - c)*(a + b - c)*x*z + 2*(a - b)*(a - c)*(a + b - c)*(a - b + c)*y*z + (a - b)^2*(a + b - c)^2*z^2 = 0

  This curve, a parabola inscribed in ABC, has these properties:

  focus: X₁₀₉

  directrix: X₁X₄

  vertex: X₆₆₉₅₇

  infinite point: X₅₂₂

  perspector: X₆₆₄

  The Moses inparabola passes through X_i for these i: 522, 3676, 4105, 39771, 43924, 57241, 57252, 58858, 58877, 62579, 66287, 66957, 66967, 66968, 66969, and 66975 to 66989.

  Let A' be the touchpoint of the parabola and line BC, and define B' and C' cyclically. The appearance of (T, i) in the following list means that the triangle A'B'C', being the cevian triangle of X₆₆₄, is perspective to T, and the perspector is X_i:

  (ABC, 664)

  (anticomplementary, 522)

  (tangential, 23865)

  (excentral, 514)

  [and othersj].

• X₆₇₀₁₀–X₆₇₀₆₆ — Points related to the 3rd Pavlov triangle

  contributed by Ivan Pavlov on Jan 27, 2025. Let I be the incenter of ABC and M_aM_bM_c its medial triangle. Let T_a, T_b, T_c be the reflections of M_a, M_b, M_c in AI, BI, CI, resp. Below, we call T_aT_bT_c the 3rd Pavlov triangle of ABC. T_aT_bT_c has the following A-vertex coordinates: {(b - c)^2, -b^2, -c^2}.

  - T_aT_bT_c is perspective to ABC with center X₆

  - T_aT_bT_c is perspective to the intouch and to the 1st Pavlov-Altintas triangles with center X₅₁₃.

  - T_aT_bT_c is perspective to the anti-Aquila and to the 2nd Pavlov triangles with center X₆₃₂₉₂.

  - T_aT_bT_c is perspective to the 5th mixtilinear with center X₃₈₄₉₆

  - T_aT_bT_c is orthologic to the 1st Ehrmann triangle with center X₆. - TaTbTcis orthologic to the intouch triangle with center X₃₂₄₄. The reciprocal orthology center is X₆₇₀₁₈.

  - The three triangles - intouch, 1st Pavlov-Altintas, and T_aT_bT_c - all have the same side-triangle, which is degenerate with veretices on the line X₁X₃.

  The inverse of T_aT_bT_c is perspective to ABC with center X₅₆₁₇₉. T_aT_bT_c is orthologic to the intouch triangle with center X₇₈ and the reciprocal orthology center is X₁₂₀₅₃.

• X₆₇₁₀₁–X₆₇₁₂₈ — Dao ortho-tripoles

  contributed by César Eliud Lozada, February 4, 2025. Let :g:`ABC << A B C >>` be a scalene triangle and ℍ a rectangular circum-hyperbola of :g:`ABC << A B C >>`, i.e., ℍ is an hyperbola through :ref:`X<sub>4</sub> <X<sub>4</sub>>` with perspector P on the orthic axis :ref:`X<sub>230</sub> <X<sub>230</sub>>`:ref:`X<sub>231</sub> <X<sub>231</sub>>` of :g:`ABC << A B C >>`. Let :g:`Q’ <Q’>`, :g:`Q″ <Q″>` be two points on ℍ(P). Denote :g:`A’ <A’>` the intersection of :g:`BC <[ B C ]>` with the perpendicular to line :g:`AQ″ <[ A Q″ ]>` through :g:`Q’ <Q’>`, and cyclically :g:`B’ <B’>`, :g:`C’ <C’>`. Similarly, denote :g:`A″ <A″>` the intersection of :g:`BC <[ B C ]>` with the perpendicular to line :g:`AQ’ <[ A Q’ ]>` through :g:`Q″ <Q″>`, and cyclically :g:`B″ <B″>`, :g:`C″ <C″>`. Then: :g:`A’ <A’>`, :g:`B’ <B’>`, :g:`C’ <C’>` are collinear on a line r’ with tripole :g:`T’ <T’>`. :g:`A″ <A″>`, :g:`B″ <B″>`, :g:`C″ <C″>` are collinear on a line r” with tripole :g:`T″ <T″>`. Lines r’ and r” are parallel. If :g:`Q’ <Q’>` = :g:`Q″ <Q″>` = Q then r’ = r” = the orthotransversal of :g:`Q <Q>`. (:person:`Dao Thanh Oai <dao-thanh-oai>` - Dec. 20, 2024)

  The point T' is named here the P-Dao ortho-tripole of-(Q’, Q”), whilst T″ is named here the P-Dao ortho-tripole of-(Q”, Q’).

  Some Dao ortho-tripoles are shown in the following table: [omittted here].

• X₆₇₂₁₂–X₆₇₂₂₂ — Points on Warren circles

  contributed by Clark Kimberling, February 12, 2025. This preamble is based on notes from Benjamin Warren, February 7, 2025. The points X₆₇₂₁₂-X₆₇₂₂₂ and X₆₇₂₂₅-X₆₇₂₃₄ were contributed by Peter Moses, February 10-12, 2025.

  In the plane of a triangle ABC with centroid G and orthocenter H, let A' be the reflection of G in the line BC, and define B' and C' cyclically. Let G_A be the midpoint of segment AA', and define G_B and G_C cyclically. The four points G, G_A, G_B, G_C lie on a circle, here named the Warren G-circle, which passes through the point X_i for these i: 2, 381, 5465, 9169, and 67212 through 67222.

  Let A″ be the reflection of H in the line BC, and define B″ and C″ cyclically. Let H_A be the midpoint of segment AA″, and define H_B and H_C cyclically. The five points H_A, H_B, H_C, X₃, X₄ lie on a circle, here named the Warren H-circle, which passes through the point X_i for these i: 3, 4, 15098, 18338, 18341, 18342, 18347, 18348, 31847, 31848, 31849, 31850, 31851, 31852, 31853, 31854, 31864, 31865, 31866, 43389, 43395, 43396, 53716, 53727, and 67225 through 67235.

  The segment X₃X₄ is a diameter of the Warren H-circle.

  The configurations for the Warren circles are partially generalized as follows: Let P be a point in the plane of ABC. Let A* be the reflection of P in BC, and define B* and C* cyclically. Let P_A be the midpoint of segment AA*, and define P_B and P_C cyclically. Then X₅ = X₃-of-P_AP_BH_P. (Benjamin Warren, February 12, 2025)

  For further properties of the Warren H-circle and an introduction to the Warren H-triangle, see the preamble just before X₇₀₂₇₀.

• X₆₇₂₄₂–X₆₇₂₆₀ — Circumconics through Morley centers

  contributed by César Eliud Lozada, February 14, 2025. This section describes perspectors of some circumconics passing through four or more Morley centers (or centers with coordinates containing trigonometric functions of thirds of angles A, B, C). A list of these centers can be seen here (MorleyCenters.html). Three already known circumconics of this kind are: The circumcircle of ABC, passing through X₁₃₅₉₃, X₁₃₅₉₄, X₆₅₃₈₀, X₆₅₃₈₂. The circumconic {{A, B, C, X₃₅₇, X₁₁₃₄, X₁₁₃₆, X₃₆₀₂, X₃₆₀₃, X₃₆₀₄, X₁₃₅₉₃, X₃₈₄₁₅, X₃₈₄₁₆, X₃₈₄₁₇, X₅₂₅₂₂}} has perspector X₅₆₃₇. The circumconic {{A, B, C, X₄, X₃₅₇, X₅₄₅₇, X₆₅₁₅₅, X₆₅₃₇₉, X₆₅₃₈₁, X₆₅₃₈₂}}, named the Hatzipolakis-Moses-Morley hyperbola in X₆₅₃₇₈, has perspector X₆₅₃₇₈.

  Conjecture: The conics in this section are all hyperbolas

• X₆₇₂₇₉–X₆₇₂₈₈ — Points on the Hagos circle

  :contributed by Clark Kimberling, based on notes by Elias M Hagos and Peter Moses, February 27, 2025. Let ABC be a triangle with orthocenter H, orthic triangle H_aH_bH_c and nine-point center N. Let A'B'C' be the anticevian triangle of N. Let A_b, A_c respectively be the projections of A' on H_aH_c and H_aH_b, and define B_c, B_a, C_a and C_b cyclically. The six points A_b, B_c, C_a, A_c, C_a, C_b lie on a circle having center X₅₁₈₈₈ and radius

  (S^6 + SA^2*SB^2*SC^2)/(8*R*S^2*SA*SB*SC),

  where R is the circumradius of ABC. See euclid 8026.

  This circle is here named the Hagos circle>, which is the Taylor-circle-inverse of the nine-point-circle-inverse of the orthic triangle. The point X_i lies on the Hagos circle for i = 67279, 67280, …, 67288.

• X₆₇₃₂₄–X₆₇₃₂₈ — Orthogonal projections on the Euler line

  contributed by Clark Kimberling and Peter Moses, March 8, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the Euler line is the point

  X₂X₃∩PX₅₂₃ = a^2*((a^2 - b^2)*(a^2 - c^2) + b^2*c^2*J^2)*p + (b^2 - c^2)*((a^2 - c^2)*(a^2 - b^2 + c^2)*q - (a^2 - b^2)*(a^2 + b^2 - c^2)*r) : : ,

  where (as in X₁₁₁₃,

  J = |OH|/R = (1/abc)[S(6) - S(2,4) + \(3a^{2}\)\(b^{2}\)\(c^{2}\)]^1/2, where S(6) = \(a^{6}\) + \(b^{6}\) + \(c^{6}\), and S(2,4) = \(a^{2}\)\(b^{4}\) + \(a^{2}\)\(c^{4}\) + \(b^{2}\)\(c^{4}\) + \(b^{2}\)\(a^{4}\) + \(c^{2}\)\(a^{4}\) + \(c^{2}\)\(b^{4}\)

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the Euler line is X_j: (1,3109), (6,1316), (75,67325), (8,36154), (9,67326), (10,36155), (11,47399), (12,47400), (13,16179), (14,16180), (15,16181), (16,16182), (19,14119), (32,36156), (35,47401), (36,47402), (39,36157), (40,36158), (42,47403), (43,47404), (49,36159), (51,34093), (52,36160), (53,57586), (54,36161), (64,36162), (69,36163), (74,36164), (75,67324), (76,36165), [and others].

• X₆₇₃₄₀–X₆₇₃₄₈ — Orthogonal projections on the Nagel line

  contributed by Clark Kimberling and Peter Moses, March 9, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the Nagel line is the point

  X₁X₂∩PX₃₆₆₇ = (2*a^3 - 4*a^2*b - a*b^2 + b^3 - 4*a^2*c + 12*a*b*c - 3*b^2*c - a*c^2 - 3*b*c^2 + c^3)*p - (a - c)*(3*a - b - c)*(b - c)*q + (a - b)*(3*a - b - c)*(b - c)*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the Nagel line is X_j: (3,6789), (4,6788), (5,67340), (7,67341), (11,67342), (20,6790), (40,67343), (63,67344), (80,67345), (84,67346), (100,67347), (104,67348), [and others].

• X₆₇₃₄₉–X₆₇₃₈₁ — Orthogonal projections on the Brocard axis

  contributed by Clark Kimberling and Peter Moses, March 10, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the Brocard axis is the point

  X₃X₆∩PX₅₁₂ = (a^2*(b^2*c^2*(2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*p + a^2*(a - c)*(b - c)*c^2*(a + c)*(b + c)*q - a^2*(a - b)*b^2*(a + b)*(b - c)*(b + c)*r) : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the Brocard axis is X_j: (1,3110), (2,3111), (4,31850), (5,67349), (13,67350), (14,67351), (20,67352), (22,67353), (23,67354), (24,67355), (25,67356), (26,67357), (31,5170), (40,67358), (49,67359), (51,15544), (55,67360), (64,67361), (69,67362), (74,67363), (76,67364), (83,67365), (98,67366), (99,67367), (101,67368), (103,67369), (106,67370), (110,9181), [and others].

• X₆₇₃₈₂–X₆₇₃₈₉ — Orthogonal projections on the line X₁X_6

  contributed by Clark Kimberling and Peter Moses, March 10, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the line X₁X₆ is the point

  X₁X₆∩PX₃₃₀₉ = a*(b*c*(2*a^4 - 4*a^3*b + 3*a^2*b^2 - 2*a*b^3 + b^4 - 4*a^3*c + 4*a^2*b*c - 2*b^3*c + 3*a^2*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4)*p + a*(a - c)*(b - c)*c*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)*q - a*(a - b)*b*(b - c)*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)*r) : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the line X₁X₆ is X_j: (3,1083), (4,67382), (5,67383), (36,67384), (40,67385), (55,67386), (57,67387), (84,67388), (100,67389), (649,41391), (650,5526), [and others].

• X₆₇₃₉₁–X₆₇₄₀₀ — Orthogonal projections on the line X₂X_6

  contributed by Clark Kimberling and Peter Moses, March 10, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the line X₁X₆ is the point

  X₂X₆∩PX₁₄₉₉ = (2*a^6 - 6*a^4*b^2 - a^2*b^4 + b^6 - 6*a^4*c^2 + 20*a^2*b^2*c^2 - 5*b^4*c^2 - a^2*c^4 - 5*b^2*c^4 + c^6)*p - (a - c)*(b - c)*(a + c)*(b + c)*(5*a^2 - b^2 - c^2)*q + (a - b)*(a + b)*(b - c)*(b + c)*(5*a^2 - b^2 - c^2)*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the line X₂X₆ is X_j: (3,5108), (4,6792), (5,32525), (20,38940), (23,67391), (25,67392), (74,67393), (98,5912), (99,56429), (110,67394), (111,50566), [and others].

• X₆₇₄₀₁–X₆₇₄₁₃ — Dao-Cevaconjugate-conics

  contributed by César Eliud Lozada, March 10, 2025. The following theorem is due to Dao Thanh Oai (March 6, 2025):

  Let ABC be a triangle and P, Q two distinct points.

  Let A_pB_pC_p be the cevian triangle of P.

  Parallel lines to AP, BP, CP through Q cut BC, CA, AB at A'_q, B'_q, C'_q, respectively.

  Let A″_q, B″_q, C″_q be the reflections of Q in A'_q, B'_q, C'_q, respectively.

  Then:

  A_pB_pC_p and A″_qB″_qC″_q are perspective, with perspector Z(P, Q)=CevaConjugate(P, Q). Points {A″_q, B″_q, C″_q, P, Q, Z(P,Q)} lie on a conic 𝒞(P, Q).

  The conic 𝒞(P, Q) is here named the P-Dao-Cevaconjugate conic of-Q.

  If P = \(x_{p}\); \(y_{p}\); \(z_{p}\) and Q = \(x_{q}\); \(y_{q}\); \(z_{q}\) (barycentrics), then

  𝒞(P, Q) = Σ( (\(y_{p}\)*\(z_{q}\)-\(y_{q}\)*\(z_{p})\)*(\(x_{p}\)*(\(y_{p}\)*\(z_{q}\)+\(y_{q}\)*\(z_{p})\)+(\(y_{q}\)+\(z_{q})\)*\(y_{p}\)*\(z_{p})\)*x^2 - \(x_{q}\)*((\(y_{p}\)-\(z_{p})\)*\(y_{p}\)*\(z_{p}\)*\(x_{q}\)+(\(x_{p}\)+\(z_{p})\)*\(x_{p}\)*\(z_{p}\)*\(y_{q}\)-(\(x_{p}\)+\(y_{p})\)*\(x_{p}\)*\(y_{p}\)*\(z_{q})\)*y*z ) = 0

  Note: 𝒞(P, Q) is a rectangular hyperbola if Q lies on the tripolar linear of the isogonal conjugate-of-the cross difference of-(P and the Orion transform of P).

  The appearance of (i, j, n) in the following list means that the center of the X_i-Dao-Cevaconjugate conic of-X_j is X_n:

  (1, 2, 67401), (1, 3, 67402), (1, 4, 67403), (1, 6, 67404), (2, 1, 4432), (2, 3, 67405), (2, 4, 67406), (2, 6, 5026), (3, 2, 67407), (3, 4, 67408), (3, 6, 67409), (4, 1, 67410), (4, 2, 12824), (4, 3, 113), (4, 6, 67411), (6, 2, 67412), (6, 3, 67413)

• X₆₇₄₁₄–X₆₇₄₅₉ — Orthogonal projections on the line X₁X_3

  contributed by Clark Kimberling and Peter Moses, March 11, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the line X₁X₆ is the point

  X₁X₃∩PX₅₁₃ = a*(b*c*(2*a^3 - 2*a^2*b - a*b^2 + b^3 - 2*a^2*c + 4*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*p + a*(a - c)*(b - c)*c*(a - b + c)*q - a*(a - b)*b*(b - c)*(a + b - c)*r) : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the line X₂X₆ is X_j: (2,34583), (4,31849), (5,67414), (6,5091), (7,67415), (8,67416), (9,67417), (10,67418), (11,14115), (12,67419), (20,67420), (21,67421), (23,67422), (24,67423), (25,67424), (31,67425), (33,67426), (34,67427), (37,67428), (38,67429), (42,67430), (43,67431), (44,1155), (45,67432), (58,67433), (59,67434), (63,67435), [and others].

• X₆₇₄₆₀–X₆₇₄₇₇ — Orthogonal projections on the line X₁X_4

  contributed by Clark Kimberling and Peter Moses, March 11, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the line X₁X₄ is the point

  X₁X₄∩PX₅₂₂ = (2*a^6 - 2*a^5*b - a^4*b^2 + 2*a^3*b^3 - 2*a^2*b^4 + b^6 - 2*a^5*c + 4*a^4*b*c - 2*a^3*b^2*c - 2*a^2*b^3*c + 4*a*b^4*c - 2*b^5*c - a^4*c^2 - 2*a^3*b*c^2 + 8*a^2*b^2*c^2 - 4*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 - 2*a^2*b*c^3 - 4*a*b^2*c^3 + 4*b^3*c^3 - 2*a^2*c^4 + 4*a*b*c^4 - b^2*c^4 - 2*b*c^5 + c^6)*p - (a - c)*(a - b - c)*(b - c)*(a - b + c)*(a^2 - b^2 + c^2)*q + (a - b)*(a - b - c)*(b - c)*(a + b - c)*(a^2 + b^2 - c^2)*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the line X₂X₆ is X_j: (2,61732), (3,31866), (5,67460), (7,67461), (8,18340), (9,67462), (10,51889), (11,67463), (20,67464), (36,67465), (40,67466), (46,67467), (55,67468), (56,67469), (57,67470), (65,67471), (78,67472), (80,67473), (100,67474), (104,67475), (109,66990), [and others].

• X₆₇₄₇₈–X₆₇₄₉₂ — Orthogonal projections on the Fermat line, X₆X_13

  contributed by Clark Kimberling and Peter Moses, March 12, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on the Fermat line, X₆X₁₃, is the point

  X₆X₁₃∩PX₆₉₀ = (2*a^10 - 4*a^8*b^2 + 4*a^6*b^4 - 3*a^4*b^6 + b^10 - 4*a^8*c^2 + 4*a^6*b^2*c^2 - a^4*b^4*c^2 + 6*a^2*b^6*c^2 - 4*b^8*c^2 + 4*a^6*c^4 - a^4*b^2*c^4 - 10*a^2*b^4*c^4 + 3*b^6*c^4 - 3*a^4*c^6 + 6*a^2*b^2*c^6 + 3*b^4*c^6 - 4*b^2*c^8 + c^10)*p - (a - c)*(b - c)*(a + c)*(b + c)*(2*a^2 - b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*q + (a - b)*(a + b)*(b - c)*(b + c)*(2*a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the Fermat line is X_j: (2,5465), (3,18332), (4,67478), (5,67479), (15,67480), (16,67481),(20,67482), (25,67483), (32,67484), (39,67485), j[and others].

• X₆₇₄₉₃–X₆₇₅₂₄ — Orthogonal projections on the anti-orthic axis, X₄₄X_513

  contributed by Clark Kimberling and Peter Moses, March 13, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₄₄X₅₁₃, is the point

  X₄₄X₅₁₃∩PX₅₁₇ = a*(b*(2*a - b - c)*(a + b - c)*c*(a - b + c)*p + a*c*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)*q + a*b*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)*r) : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the X₄₄X₅₁₃ is X_j:

  (1,1155), (2,67493), (4,67494), (5,38472), (6,67495), (9,67496), (11,67497), (19,67498), (20,67499), (31,67500), [and others].

• X₆₇₅₂₅–X₆₇₅₂₈ — Orthogonal projections on the orthic axis, X₂₃₀X_231

  contributed by Clark Kimberling and Peter Moses, March 13, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₂₃₀X₂₃₁, is the point

  X₂₃₀X₂₃₁∩PX₃₀ = (2*a^6 - a^4*b^2 - 4*a^2*b^4 + 3*b^6 - a^4*c^2 + 8*a^2*b^2*c^2 - 3*b^4*c^2 - 4*a^2*c^4 - 3*b^2*c^4 + 3*c^6)*p + (a^2 - b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*q + (a^2 + b^2 - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the X₂₃₀X₂₃₁ is X_j: (1,16272), (2,468), (6,16303), (8,16304), (10,16305), (11,47140), (12,47160), (13,47141), (14,47142), (19,47161), (32,16306), (37,16307), (39,16308), (40,16309), (50,16310), (51,16311), (52,47143), (53,47144), (54,47145), (58,47163), (64,47164), (67,47165), (69,16312), (74,47146), (76,16313), (83,16314), (98,16315), (99,16316), (107,47147), (110,47148), [and others].

• X₆₇₅₂₉ — Intercepts of the line through X₄ and orthogonal to the Euler line

  contributed by Clark Kimberling and Peter Moses, March 13, 2025. Let L be the line through X₄ and orthogonal to the Euler line. The appearance of (line,X_iX_j, k) in the following list means that the line X_iX_j intercepts L in X_k:

  (Brocard axis, X₃X₆, 67529)

  (Nagel line, X₁X₂, 67530)

  (X₂X₆, 67531)

  (Fermat line, X₆X₁₃, 66167)

  (X₁X₃, 67532)

  (anti-orthic axis, X₄₄X₅₁₃₃, 67533)

  (Lemoine axis, X₁₈₇X₂₃₇, 67534)

  (X₁X₅, 67535)

• X₆₇₅₃₉–X₆₇₅₆₄ — Orthogonal projections on the Lemoine axis, X₁₈₇X_237

  contributed by Clark Kimberling and Peter Moses, March 14, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₁₈₇X₂₃₇, is the point

  a^2*(b^2*c^2*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*p + a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(c^2*q + b^2*r)) : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the X₁₈₇X₂₃₇ is X_j: (1,63822), (2,47638), (3,187), (4,5167), (5,67539), (20,67540), (22,67541), (23,1495), (24,67542), (25,67543), (26,67544), (31,67545), (40,67546), (42,67547), (55,67548), (74,67549), (98,67550), (99,67551), (101,67552), (111,67553), (112,67554), [and others].

• X₆₇₅₆₇–X₆₇₅₈₃ — Orthogonal projections on the Soddy line, X₁X_7

  contributed by Clark Kimberling and Peter Moses, March 14, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₁X₇, is the point

  X₁X₇∩PX₅₁₄ = (2*a^4 - 2*a^3*b - a^2*b^2 + b^4 - 2*a^3*c + 4*a^2*b*c - 2*b^3*c - a^2*c^2 + 2*b^2*c^2 - 2*b*c^3 + c^4)*p - (a - c)*(-b + c)*(a - b + c)^2*q - (a - b)*(b - c)*(a + b - c)^2*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on the X₁X₇ is X_j: (2,38941), (3,67567), (4,67568), (5,67569), (8,67570), (9,67571), (10,67572), (36,67573), (40,67574), (55,67575), (57,67576), (65,67577), (69,67578), (85,67579), (101,67580), [and others].

• X₆₇₅₈₄–X₆₇₅₉₅ — Points associated with circles tangent to the Euler line

  contributed by Clark Kimberling and Peter Moses, March 15, 2025. This section presents points on two circles that are tangent to the Euler line at X₂:

  The circle with center X₁₆₄₉ passes through X_i for i = 2, 110, 2770, 5463, 5464, 7998, 9168, 9829, 10717, 14916, 34312, 52722, 52723, 67584, 67585, 67586, 67587.

  The circle with center X₆₇₅₈₈ passes through X_i for i = 2, 126, 618, 619, 1649, 3258, 5108, 5642, 5650, 7711, 10163, 14685, 67589, 67590, 67591, 67592, 67593, 67594, 67595, [and others].

• X₆₇₅₉₆–X₆₇₆₂₃ — Orthogonal projections on the de Longchamps Axis, X₃₂₅X_523

  contributed by Clark Kimberling and Peter Moses, March 15, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₃₂₅X₅₂₃, is the point

  X₃₂₅X₅₂₃∩PX₃₀ = (a^4*b^2 + a^2*b^4 - 2*b^6 + a^4*c^2 - 4*a^2*b^2*c^2 + 2*b^4*c^2 + a^2*c^4 + 2*b^2*c^4 - 2*c^6)*p - b^2*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*q - c^2*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on X₃₂₅X₅₂₃ is X_j: (1,67596), (2,858}, (6,67597}, (8,50144}, (10,67598}, (32,67599}, (39,67600}, (40,67601}, (64,67602}, (69,67603}, (74,67604}, (75,67605}, (76,67606}, (98,67607}, (99,325}, (100,67608}, (104,67609}, (107,67610}, (110,67611}, (111,67612}, (112,67613}, [and others].

• X₆₇₆₂₄–X₆₇₆₄₂ — Points associated with the Warren reflection triangle

  contributed by Clark Kimberling and Peter Moses, based on notes from Benjamin Warren, March 13-16, 2025. In the plane of a triangle ABC, let A' be the reflection of A in X₂, and let A″ be the reflection of A' in line BC. Define B″ and B″ cyclically. The triangle A″B″C is here named the Warren reflection triangle, and its circumcircle, the Warren reflection circle. The vertices are given by

  A″ = a^2 : c^2 + a^2 - b^2 : a^2 + b^2 - c^2

  B″ = b^2 + c^2 - a^2 : b^2 : a^2 + b^2 - c^2

  C″ = b^2 + c^2 - a^2 : c^2 + a^2 - b^2: c^2

  Let A’’’B’’’C’’’ be the reflection of A″B″C″ in X₃. Then

  A’’’ = a^2 : c^2 + a^2 - b^2 : a^2 + b^2 - c^2

  B’’’ = b^2 + c^2 - a^2 : b^2 : a^2 + b^2 - c^2

  C’’’ = b^2 + c^2 - a^2 : c^2 + a^2 - b^2: c^2
  A’’’ = -(a^2*(5*a^4 - 4*a^2*b^2 - b^4 - 4*a^2*c^2 + 2*b^2*c^2 - c^4)) : (a^2 - b^2 + c^2)*(a^4 + 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : (a^2 + b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + 4*a^2*c^2 - 2*b^2*c^2 + c^4),

  and B’’’ and C’’’ are given cyclically.

  Along with the six points A″, B″, C″, A’’’, B’’’,C’’’ on the Warren reflection circle are the points X_i for these i: 2, 376,1106, 14916, 67625 through 67642; at total of 24 points. The Warren reflection circle has persepctor X₆₇₆₂₄ and is given by the following equation:

  c^2*x*y + b^2*x*z + a^2*y*z

  - (x + y + z)*(((a^2 + b^2 + c^2)*x)/9

  + ((a^2 + b^2 + c^2)*y)/9

  + ((a^2 + b^2 + c^2)*z)/9)

  = 0.

  The appearance of (name, i) in the following list means that the Warren reflection triangle is perspective to the named triangle,a nd the perspector is X_i:

  (ABC, 69)

  (tangential triangle, 2916)

  (reflection triangle of ABC in X₃, 4549

  (2nd Brocard triangle (CTC), 141)

  (orthocentroidal triangle (see ETC X₅₄₇₆), 2)

  [and others].

  Returning to the Warren reflection triangle, the point, let A”” be the reflection of A″ in X₂, and define B”” and C”” cyclically. Then

  A”” = a^2 : a^2 + b^2 - c^2 : a^2 + c^2 - b^2

  B”” = b^2 + a^2 - c^2 : b^2 : b^2 + c^2 - a^2

  C”” = c^2 + a^2 - b^2 : c^2 + b^2 - a^2 : c^2

  The vertices A””, B””, C”” lie on the orthocentroidal circle of ABC.

  For every point P on the cubic K060, A””B””C”” is perspective to the cevian triangle of P,

  For every point P on the cubic K001, A””B””C”” is perspective to the anticevian triangle of P,

  For every point P on the Keipert circumhyperbola, A””B””C”” is perspective to the pedal triangle of P.

  For every point P on the cubic K1279, A””B””C”” is perspective to the antipedal triangle of P.

  The appearance of (name, i) in the following list means that the triangle A””B””C”” is perspective to the named triangle, and the perspector is X_i:

  (ABC, 4)

  (orthic triangle, 4)

  (excentral triangle, 3336)

  (Euler triangle, 4

  [and others].

• X₆₇₆₄₃–X₆₇₆₆₁ — Orthogonal projections on the Gergonne line, X₂₄₁X_514

  contributed by Clark Kimberling and Peter Moses, March 16, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₂₄₁X₅₁₄ is the point

  X₂₄₁X₅₁₄∩PX_5165 = 2*a^4 - a^3*b - a^2*b^2 - 3*a*b^3 + 3*b^4 - a^3*c + 2*a^2*b*c + 3*a*b^2*c - 4*b^3*c - a^2*c^2 + 3*a*b*c^2 + 2*b^2*c^2 - 3*a*c^3 - 4*b*c^3 + 3*c^4)*p + (a - b + c)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3)*q + (a + b - c)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3)*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on X₃₂₅X₅₂₃ is X_j: (1,1323), (2,67643), (3,51775), (4,8074), (5,67644), (6,67645), (8,67646), (11,3911), (36,67647), (37,67648), (44,67649), (46,67650), (55,67651), (56,67652), (57,67653), (65,67654), (72,67655), (80,67656), (100,67657), [and others].

• X₆₇₆₆₂–X₆₇₆₇₃ — Orthogonal projections on the van Aubel line, X₄X_6

  contributed by Clark Kimberling and Peter Moses, March 16, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₄X₆ is the point

  X₄X₆∩PX₅₂₅ = (2*a^10 - 2*a^8*b^2 + a^6*b^4 - 3*a^4*b^6 + a^2*b^8 + b^10 - 2*a^8*c^2 + 3*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - 3*b^8*c^2 + a^6*c^4 + 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 2*b^6*c^4 - 3*a^4*c^6 + 2*a^2*b^2*c^6 + 2*b^4*c^6 + a^2*c^8 - 3*b^2*c^8 + c^10)*p - (a - c)*(b - c)*(a + c)*(b + c)*(a^2 - b^2 - c^2)*(a^2 - b^2 + c^2)^2*q + (a - b)*(a + b)*(b - c)*(b + c)*(a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)^2*r : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on X₄X₆ is X_j: (2,6794), (3,18338), (5,43278), (20,67662), (25,67663), (52,67664), (68,67665), (69,35902), (76,67666), (99,67667), (110,67668), [and others].

• X₆₇₆₈₅–X₆₇₆₉₅ — Orthogonal projections on the Steiner minor axis, X₁X(X₃₄₁₄

  contributed by Clark Kimberling and Peter Moses, March 18, 2025. The orthogonal projection of a point P =\(p:q:r\) (barycentrics) on X₁X(X₃₄₁₄ is the point

  ((2*a^2 - b^2 - c^2)*p - (a^2 + b^2 - 2*c^2)*q - (a^2 - 2*b^2 + c^2)*r + \(\sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}\)*(4*p + q + r)) : :

  The appearance of (i,j) in the following list means that the orthogonal projection of X_i on X₁X(X₃₄₁₄ is X_j: (1,67685), (3,47089), (4,31862), (5,67686), (20,51899), (69,67687), (76,67688), (83,67689), (98,67690), (99,66625), (110,67691), [and others].

• X₆₇₇₁₄–X₆₇₈₄₂ — Points on Warren reflection circles

  contributed by Clark Kimberling and Peter Moses, March 26, 2025. As a generalization of the triangles and circles introduced by Benjamin Warren, as in the preamble just before X₆₇₆₂₄), let A' be the reflection of A in a point P =\(p : q : r\), let A″ be the reflection of A' in line BC, and define B″ and C″ cyclically. Then

  A″ = a^2(q + r - p)

  : (a^2 + b^2 - c^2)(p - r) + (a^2 - b^2 + c^2) q

  : (a^2 - b^2 + c^2)(p - q) + (a^2 + b^2 - c^2) r

  Denote the triangle A″B″C″ as WT(P) and its circumcircle as WC(P). The concentric circles WC(P) are listed here for selected points P. Specifically, the appearance of

  {h, {i(1), i(2), … i(n)}}

  means that the points X(i(1)), X(i(2)), …, X(i(n)) lie on the circle XC(X(h)).

  {1, {8, 944, 18340, 38497, 38498, 38499, 38500, 38501, 38502, 38503, 38504, 38505, 38506, 38507, 38508, 38509, 38510, 38511, 38512, 38513, 38514, 38515, 38516, 38517, 38518, 38519, 67714, 67715, 67716}}

  ——————————-

  {2, {2, 376, 11006, 14916, 67625, 67626, 67627, 67628, 67629, 67630, 67631, 67632, 67633, 67634, 67635, 67636, 67637, 67638, 67639, 67640, 67641, 67642, 67717, 67718, 67719, 67720}}

  ——————————-

  {3, {4, 20, 18337, 18339, 32616, 32617, 67464, 67568, 67662, 67721}}

  ——————————-

  {10, {1, 40, 13534, 22939, 47270, 67343, 67385, 67466, 67574, 67722, 67723, 67724, 67725, 67726}}

  ——————————-

  {11, {74, 98, 99, …., [many others]}} This is the circumcircle.

  ——————————-

  {12, {2975, 11491, 38555, 38556, 38557, 38558, 38559, 38560, 38561, 38562, 38563, 38564, 38565, 38566, 38567, 38568, 38569, 38570, 38571, 67811, 67812, 67813, 67814, 67815}}

  ——————————-

  [and others]

  The points X₆₇₇₁₄ - X₆₇₈₉₂ are reflections in X₃ of points on the circles WC(X_k). On such a circle, if {U, U'} and {V, V'} are pairs of antipodes, then UVU’V' is a rectangle inscribed in the circle. The appearance in the next list of

  {h, {{i(1), j(1)}, {i(2),j(2)}, …, {i(n),j(n)}}}

  means that {X(i(k)), X(i(j))} is an antipodal pair of points on WC(X(h)).

  {1, {{8, 944}, {38497, 38508}, {38498, 38499}, {38500, 38502}, {38501, 38507}, {38504, 38515}, {38505, 38516}, {38506, 38517}, {38509, 38518}, {38510, 38519}, {38512, 38513}}}

  ————————-

  {2, {{2, 376}, {11006, 67641}, {67625, 67632}, {67627, 67634}, {67628, 67635}, {67630, 67639}, {67636, 67640}}}

  ————————-

  {3, {{4, 20}, {18337, 67662}, {18339, 67464}, {32616, 32617}}}

  ————————-

  {10, {{1, 40}}}

  ————————-

  {11, {{74, 110}, {98, 99}, {100, 104}, {101, 103}, {102, 109}, {105, 1292}, {106, 1293}, {107, 1294}, {108, 1295}, {111, 1296}, {112, 1297}, {476, 477}, [and many others] ————————-

  {12, {{2975, 11491}, {38555, 38566}, {38556, 38557}, {38558, 38560}, {38559, 38565}, {38567, 38571}, {38568, 38569}}}

  ————————-

  {39, {{76, 11257}, {13325, 13326}, {38520, 38523}, {38525, 38529}, {38526, 38528}}}

  ————————-

  [and others]

• X₆₇₈₄₄–X₆₈₀₆₁ — Points related to CTR28 triangles

  contributed by Ivan Pavlov on March 25, 2025. If P_aP_bP_c is the cevian triangle of P, CTR28(P) is the triangle with vertices at the orthocenters of APbPc, BPaPc, and CPaPb. The points in this section are related to P = X₁, X₂, X₃, X₄, X₇, X₈, X₂₀, X₆₉, X₁₈₉, X₂₅₃, X₃₂₉. Note that:

  - CTR28-2, i.e when P=X₂, is the Euler traingle.

  - CTR28-4, i.e when P=X₄, is the reflection of the orthic triangle in X₃₈₉.

  - CTR28-7, i.e when P=X₇, is the reflection of the intouch triangle in X₉₄₂.

  - CTR28-8, i.e when P=X₈, is the reflection of the extouch triangle in X₅₇₇₇.

  - CTR28-69, i.e when P=X₆₉, is homothetic to the midheight triangle with center X₁₈₅.

  In general, CTR28(P) is the reflection of the cevian trianlgle of P, whenever P lies on the Lucas cubic K007. In this case it is also bilogic to ABC.

  Some of the properties in this section refer to CTR-triangles. More info on these series of triangles is available in this catalog. Also, the abbreviation UCFT means unary cofactor triangle.

Part 35

• X₆₈₀₈₆–X₆₈₀₈₉ — Points on the Moses HK-parabola

  contributed by Clark Kimberling, based on notes and data from Peter Moses, March 26, 2025. The Moses HK-parabola is introduced here as the inscribed parabola having focus X₁₁₂ and directrix the line HK = X₄X₆. The Moses HK-parabola passes through X_i for these i: 525, 2501, 14401, 15639, 17925, 17926, 23090, 32320, 43925, 52131, 52132, 57195, 57201, 57202, 57203, 57204, 58760, 58780, 58812, 60505, 68086, 68087, 68088, 68089

• X₆₈₁₀₁–X₆₈₁₂₅ — Points on the Moses X₄X_8-parabola

  contributed by Clark Kimberling, based on notes and data from Peter Moses, March 29, 2025. The Moses X₄X₈-parabola is introduced here as the parabola having focus X₁₀₀ and directrix the line HK = X₄X₈. The Moses X₄X₈-parabola passes through X_i for these i: 513, 693, 4036, 4397, 14434, 15632, 25142, 27855, 50487, 62430, and 68101–68125.

• X₆₈₁₇₉–X₆₈₂₃₂ — Harmonic pencil and harmonic lines

  contributed by César Eliud Lozada, April 3, 2025. Let \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\) be four distinct lines concurrent in a point P. These four lines are said to be an harmonic pencil (or harmonic bundle) if there exists a line ρ, intersecting them at Q₁, Q₂, Q₃, Q₄, respectively, and such that (Q₁, Q₂; Q₃, Q₄) are in harmonic range.

  The most important fact in the above definition is that if such harmonic range occurs for a line intersecting the pencil of lines, it occurs for any other line intersecting that pencil.

  In this section, three concurrent lines \(r_{1}\), \(r_{2}\), \(r_{3}\) are given and the tripole of the fourth line \(r_{4}\) is calculated, such that \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\) form an harmonic pencil. This fourth line \(r_{4}\) is denoted here as the {\(r_{1}\), \(r_{2}\)}-harmonic line of-\(r_{3}\).

  The appearance of (\(r_{1}\), \(r_{2}\), \(r_{3})\)→n in the following lists means that the tripole of the {\(r_{1}\), \(r_{2}\)}-harmonic line of-\(r_{3}\) is X_n: [lists omitted here].

  Note: These results come from a specific application of a more general theorem involving the conservation of cross-ratios. For more information, see this link.

• X₆₈₂₆₈–X₆₈₃₂₄ — Co-normal points and co-normals hyperbolas

  contributed by César Eliud Lozada, April 11, 2025. Let 𝒞 be a conic (not a circle) and 𝒩 a point on the plane of 𝒞 and not on it. The points on 𝒞 whose normals concur at 𝒩 are called the 𝒩-co-normal points of-𝒞.

• X₆₈₃₂₆–X₆₈₃₃₃ — Points on Warren Q-circles

  based on notes from Benjamin Warren, April 10, 2025. The points X₆₈₃₂₆-X₆₈₃₃₃ and list of Q-circles were contributed by Peter Moses, April 18, 2025. In the plane of a triangle ABC, let P be a point. Let

  A'B'C' = circumcevian triangle of P

  M_a = midpoint of segment BC

  A″ = reflection of A' in M_a, and define B″ and C″ cyclically

  P' = anticomplement of P.

  The circle {{A″,B″,C″}}, here introduced as the Warren Q-circle, has diameter P'X₄. See the note at X₄.

  The appearance of (n, (name), {i(1), i(2),…, i(k)}, m) in the following list means that if X_n = Q, then the points X(i(1)), X(i(2)),…, X(i(k)) lie on the circle, which has center X(m).

  (2, (orthocentroidal circle), {2, 4, 6235, 6324, 6785, 6787, 6788, 6792, 6794, 8426, 8427, 9144, 10773, 11005, 13522, 13524, 13531, 14700, 15924, 22540, 31862, 31863, 34235, 61729, 61730, 61731, 61732, 67224}, 381)

  (3, {3, 4, 15098, 18338, 18341, 18342, 18347, 18348, 31847, 31848, 31849, 31850, 31851, 31852, 31853, 31854, 31864, 31865, 31866, 43389, 43395, 43396, 53716, 53727, 67225, 67226, 67227, 67228, 67229, 67230, 67231, 67232, 67233, 67234}, 5)

  (6, (orthosymmedial circle), {4, 6, 1316, 6792, 12508, 13239, 23322, 31850, 52465, 52466, 52471, 67382, 67478}, 5480)

  (8, (Fuhrmann circle), {4, 8, 6788, 10774, 13498, 13514, 13543, 13545, 13547, 13549, 18328, 18339, 18340, 18341, 18343, 18865, 36154}, 355)

  (20, {4, 20, 18337, 18339, 32616, 32617, 67464, 67568, 67662, 67721}, 3)

  (69, {4, 69, 6792, 18331, 18335, 18337, 18343, 18347, 35902, 36163}, 1352)

  (99, {4, 99, 112, 7472, 12833, 15342, 18331, 46046, 67224, 67232, 67667}, 114)

  (100, {4, 100, 108, 10773, 15343, 18341, 34151, 36167, 46044, 67474}, 119) [and others].

• X₆₈₃₃₄–X₆₈₃₅₅ — Antiproducts

  contributed by Ivan Pavlov on Apr 22, 2025. If P=(\(u:v:w\)) and Q=(\(p:q:r\)) in barycentric coordinates, we define the antiproduct of P and Q as the point (-p u + q v + r w : p u - q v + r w : p u + q v - r w).

  The antiproduct of two points can be conveniently constructed as the anticomplement of their barycentric product. Note that the antiproduct of P and the isogonal conjugate of P is always X₆₉. The antiproduct of P and the isotomic conjugate of P is always X₂. Of course, the antiproduct of any point P and the centroid is the anticomplement of P.

  The following relations also holds:

  (1) II-Caph point of P = antiproduct of complement and anticomplement of P. See X₃₂₀₀₁ for definition of II-Caph.

  (2) Vijay 6th parallel transform of P = antiproduct of complement and isotomic conjugate of P.

  (3) M(P) = antiproduct of G and (KP₂(G) of P and P), see the preambles of X₄₀₈₉₆ and X₅₅₉₁₇ for definitions of M and KP₂. G denotes the centroid.

  (4) The antiproduct of the crosspoint and the cevian product of P and Q coincides with the antiproduct of P and Q.

  (5) The antiproduct of G and the infinity point of the tripolar of G coincides with the perspector of the conic {A, B, C,G,P}.

• X₆₈₃₅₆–X₆₈₃₆₉ — Similar inscribed & circumscribed triangles

  contributed by César Eliud Lozada, April 22, 2025.

  (1) Given a triangle ABC, to find points A', B', C' such that A'B'C' is inscribed in and directly similar to ABC.

  2. Given a triangle ABC, to find points A″, B″, C″ such that A″B″C″ is circumscribed to and directly similar to ABC.

• X₆₈₃₇₅–X₆₈₃₈₅ — Points on with the Warren G-circles

  contributed by Clark Kimberling and Peter Moses, based on notes from Benjamin Warren, April 22, 2025.

  Let P be a point in the plane of a triangle ABC, and let

  A'B'C' = medial triangle

  A_b = reflection of A in line B'P, and define B_c and C_a cyclically

  A_c = reflection of A in line C'P, and define B_a and C_b cyclically

  P_a = circumcenter of AB_cC_b, and definer P_b and P_c cyclically.

  The centroids of the following four points lie on a circle here named the Warren G(P)-circle: ABC, APbPc, BPcPa, CPaPb.

  If P =\(p:q:r\), then barycentrics for the center of the Warren G(P)-circle are given by [long expression omtted here].

  For example, the center of the Warren G(X₁₀)-circle is X₅₈₈₆, and the squared radius is R*(R - 2*r) / 9.

  The appearance of i in the following list of 16 points means that X_i lies on the Warren G(X₁₀)-circle: 2, 5603, 32631, 61732, 67625, and 68375, 68376, …, 68385.

• X₆₈₅₆₁–X₆₈₅₈₁ — Points on the Huygens hyperbola

  contributed by Clark Kimberling, based on notes and data from Peter Moses, May 24, 2025. The Hugens hyperbola is the isogonal conjugate of the line X₃X₄₉. See Alperin, Roger C., “The Poncelet Pencil of Rectangular Hyperbolas”, :cite:`Forum Geometricorum <forum-geometricorum>`, 10 (2010), 15-20: <the journal, archived and Sandor Nagydobai Kiss, “The Poncelet Pencil’s Hyperbolas as Locus Geometric and Their Equations in Barycentric Coordinates”: article. The Huygens hyperbola, given by the barycentric equation

  (a^2 - b^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*x*y + (a^2 + b^2 - c^2)*(-a^2 + c^2)*(-a^2 + b^2 + c^2)*x*z + (b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*y*z = 0,

  has center X₁₃₆ and perspector X₂₅₀₁. It passes through X_i for these i: 4, 93, 225, 254, 264, 393, 847, 1093, 1105, 1179, 1217, 1300, 1826, 6344, 6526, 6531, 8737, 8738, 8741, 8742, 8801, 8884, 14860, 15424, 16263, 17983, 18808, 18846, 18847, 18848, 18849, 18850, 18851, 18852, 18853, 18854, 18855, 24243, 24244, 32085, 34208, 35142, 36611, 36612, 38427, 38428, 40402, 41013, 41515, 41516, 42377, 47735, 52487, 55031, 55972, 56340, 57931, 59278, 60836, 64844, 66596, and X₆₈₅₆₁-to-X₆₈₅₈₁.

• X₆₉₀₉₁–X₆₉₀₉₈ — Additive associates

  contributed by Clark Kimberling and Peter Moses, June 18-30, 2025. Suppose that X is a triangle center, given by barycentrics \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where f(a,b,c) is a sum of n terms t(i) each of the form k a^p b^q c^r, where k is a nonzero constant, n>=1, and p,q,r are real numbers. Referring to the representation

  (*) \(f(a,b,c) = + k(1) a^p(1) b^q(1) c^r(1) +\)… + k(n) a^p(n) b^q(n) c^r(n),

  note that f(a,b,c) is a polynomial in a,b,c such that, in accord with the definition of triangle center, \(f(a,b,c) = f(b\),c,a) = f(c,a,b) and \(f(a,b,c) = f(a\),c,b), so that X is, in the literature, a polynomial triangle center, abbreviated as PTC.

  In (*), the symbol + occurs n times, for which we write the n-tuple (+,+,…+). By varying the symbols so that the first + remains fixed, and the others can be +, 0, or - , or equivalently, 1, 0, or -1, we obtain 3^(n-1) n-tuples, and a corresponding set of 3^(n-1) points by varying the additive symbols in (*), by which we mean coefficient 1, 0, or -1. The resulting family of n-tuples, or points, are here named primary additive associates (AAs) of X. These points are of two kinds: triangle centers and bicentric pairs.

  Let S(X) denote the set of AA’s of X. If X' and X″ are distinct PTC’s points in S(X), then S(X) possibly also contains a pair of (X',X″)-harmonic conjugates, specifically, the sum and difference of X' and X″, which we denote by X' + X″ and X' - X″. Either these are PTC’s, or else they are a bicentric pair. See Example 3 below.

  Example 1. Represent the b^2 + c^2 + b c as (1,1,1). The AA for (1,1,-1) is the PTC b^2 - c^2 + b c : : . On the other hand, (1,-1,1) and (1,-1,-1) represent the bicentric pair of points, b^2 - c^2 + b c : : and b^2 - c^2 - b c.

  In most of the rest of this preamble, AA’s are understood to represent triangle centers, not bicentric points.

  Example 2. Taking \(f(a,b,c) = a^{6} - a^{4} b^{2} - a^{4} c^{2} + a^{2} b^{2} c^{2}\), so that X = X₁₁₀, the family of AA’s of X has ten PTC’s, as follows:

  Additive associates of X₁₁₀ X₁₁₀ (1,1,1,1) X₅₀₁₂ (1,1,1,-1) X₂₅₁ (1,-1,-1,1) X₁₆₂₇ (1,-1,-1,-1) X₁₈₄ (1,1,1,0) X₁₉₁₅ (1,0,0,1) X₁₅₀₁ (1,0,0,0) X₂ (0,0,0,1) X₃₀₅₁ (0,1,1,0) X₁₆₁₃ (0,1,1,1)

  In the following tables, the AA consisting of all 1’s has terms in the order given by the Mathematica command Expand, as in Examples 1 and 2. [tables omitted here].

  Example 3. Continuing the above discussion of harmonic conjugates, There are C(53,2) = 1378 pairs of points X', X″ that can be chosen from the 53 additive associates of X₁₁. One such choice is X' = X₁₄₁ and X″=X₃₇₀₃, with respect to which X₃₈ and X₁₅₅₂₃ are harmonic conjugates. To verify this, it is expedient to use the equals symbol, =, between pairs of 7-tuples that represent the same point. Then

  X' = X₁₄₁ = (1,-1,0,1,1,1,-1)

  X″ = X₃₇₀₃ = (1,1,0,-1,1,-1,1)

  X' + X″ = (2,0,0,0,2,0,0) = (1,0,0,0,1,0,0) = X₃₈

  X' - X″ = (0,-2,0,2,0,2,-2) = (0,1,0 -1,0,-1,1) = X₁₅₅₂₃

  To illustrate the possiblity that for other choices of X' and X″, the harmonic conjugates are a bicentric pair, we have

  X' = X₃₂₁ = (0,0,0,1,0,1,0)

  X″ = X₆₉₃ = (0,0,0,1,0,-1,0)

  X' + X″ = (0,0,0,2,0,0,0) = (0,0,0,1,0,0,0), which represents b^2 c : c^2 a : a^2 b

  X' - X″ = (0,0,0,0,0,2,0) = (0,0,0,0,0,1,0), which represents b c^2 : c a^2 : a b^2

  Note that the bicentric pair, (X' + X″, X' - X″) = (a/b : : , a/c : : ).

  For some choices of X' and X″, at least one of the harrnonic conjugates is a primary additive associate, while the other is a more general additive associate (not formally defined above). For example,

  X'= X₁₁ = (1,1,1,1,1,1,1)

  X″ = X₅₂₃ = (1,-1,0,1,-1,-1,1)

  X' + X″ = (2,1,1,1,2,1,1) = 2 a^b^2 - b^3 - 2 a b c + b^2 c + 2 a c^2 + b c^2 - c^3 : :

  X' - X″ = (0,1,1,1,0,1,1) = X₆₉₁₇₃.

  The above discussion of AA’s is based on the Mathematica order in which the terms of the first barycentric of a point are arranged. The set of these AA’s include not only polynomial triangle centers, but also bicentric pairs. For each degree n of a PTC, there is a natural way to represent all PTC’s of that degree, without bicentric pairs. We begin with collections of elements, denoted by E(n), and defined inductively as follows:

  E(0) = {1}

  E(1) = {b + c}

  E(n) = {b^n + c^2}∪ b c E(n-2), for n>=2.
  E(2) = {b^2 + c^2, b c}

  E(3) = {b^3 + c^3, b c (b + c)}

  E(4) = {b^4 + c^4, b c (b^2 + c^2), b^2 c^2)}.

  It is easy to find that the cardinality of E(n) is [(n+2)/(2)], where [ ] = floor.

  In general, each even PTC of degree n has a representation

  h(1) a^n + h(2) a^(n-1) p(1,b,c) + h(3) a^(n-2) p(2,b,c) + … ,

  where each p(i,b,c) is a linear combination of the elements in E(n-2 i), and h(i) is a constant. The trivial linear combination, given by \(h(i)=0\) for all i, does not represent a triangle center. The odd PTC’s, as defined elsewhere in the literature, are simply those of the form (b - c)*e(a,b,c), where e(a,b,c) is an even PTC. Thus, every odd PTC of degree n+1 has a representation

  (b - c)(h(1) a^n + h(2) a^(n-1) p(1,b,c) + h(3) a^(n-2) p(2,b,c) + … ).

  While this manner of expressing PTC’s is conceptually natural, it nevertheless leads to a conclusion that various sets of barycentric products and isotomic conjugates, etc., can be highly elaborate and difficult to enumerate.

  For a further discussion of additive associates, see the preamble just before X₆₉₃₀₂.

• X₆₉₃₀₂–X₆₉₃₇₆ — Additive associates

  contributed by Clark Kimberling and Peter Moses, July 4, 2025. Continuing the discussion of odd polynomial triangle centers in the preamble just before X₆₉₀₉₁, every odd polynomial center of degree 4 has a representation

  (b-c)(k1*a^3 + k2*a^2 (b+c) + k3*a (b^2+c^2) + k4 a b c + k5 (b*3+c^3) + k6 b c (b+c)),

  which we abbreviate as ODD<k1,k2,k3,k4,k5,k6>. Here, the use of angle brackets, < and >, distinguishes this type of representation from the additive associate notation introduced in the preamble just before X₆₉₀₉₁, which uses parentheses and depend on Mathematica ordering of terms instead of elements E(n) as defined near X₆₉₀₉₁. (Of course, the latter are additive associates, too, but are restricted to triangle centers, whereas in general, additive associates include bicentric pairs of points.)

  Centers X₆₉₃₀₂-X₆₉₃₇₆ are polynomial triangle centers of degree 4, in addition to centers listed below with indexes < 69302. The appearance of

  ODD<k1,k2,k3,k4,k5,k6>, n

  in the list means that X_n is the center given by ODD<k1,k2,k3,k4,k5,k6>. If H and K are two such distinct centers, then H+K and H-K are also of this type, where + an - are the usual vector addition and subtraction. The centers, H, K, H+K, H-K comprise a harmonic range. In many cases, if all the coefficients in the representations for H and K are in the set {-1,0,1}, then the coefficients for H+K and H-K are also in the set {-1,0,1}.

  ODD<1,1,1,1,1,1>, 514

  ODD<1,1,1,1,0,0>, 513

  ODD<1,1,1,0,1,1>, 514

  ODD<1,1,1,0,0,-1>, 24719

  ODD<1,1,1,-1,1,1>, 514

  [and many others].
  Center X₆₉₄₆₁, Center of the Warren six-circle,contributed by Clark Kimberling, September 3, 2025. This preamble is based on notes from Benjamin Warren, September 2, 2025. Barycentric coordinates and equations were found by Peter Moses, September 3, 2025. In the plane of a triangle ABC, let

  I = incenter = X₁;

  P_a= parabola having focus I and directrix the line BC, and define P_b and P_c cyclically;

  A' = the point of intersection of P_b and P_c inside triangle ABC, and define B' and C' cyclically;

  T_b = line tangent to parabola P_a at B';

  T_c = line tangent to parabola P_a at C';

  A″ = T_b&capTc, and define B″ and C″ cyclically.

  The circumcenters of the six triangles A'B″I, B'C″I, C'A″I, A'C″I, B'A″I, C'B″I are concyclic.

  (Benjamin Warren, September 2, 2025) The circle is here named the Warren six-circle.

  Peter Moses found that an equation for the parabola P_a is the following:

  (a - b - c)*(a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 2*a^3*c + 2*a*b^2*c + 2*a*b*c^2 + 2*b^2*c^2 - 2*a*c^3 - c^4)*x^2 + 4*a^2*(a - b - c)*c*(a - b + c)*x*y + 4*a^3*c*(a - b + c)*y^2 + 4*a^2*b*(a - b - c)*(a + b - c)*x*z - 4*a^3*(a + b - c)*(a - b + c)*y*z + 4*a^3*b*(a + b - c)*z^2 = 0.

  Equations for P_b and P_c are determined cyclically. Related points are given by [barycentrics omited here].

Part 36

• X₇₀₂₃₁–X₇₀₂₃₆ — Points associated with P-Brocard triangles

  contributed by Peter Moses and Clark Kimberling, October 17, 2025. The P-Brocard triangle is defined at X₅₆₄₂ as follows: Let O = X₃ and suppose that P is a point other than O. Let OP be the circle with segment PO as diameter. Let A' be the point of intersection, other than O, of OP and the perpendicular bisector of segment BC, and define B' and C' cyclically. Triangle A'B'C' is called the P-Brocard triangle, and X₅₆₄₂ is X₂₃-of-the-X₂-Brocard triangle. (Randy Hutson, June 16-17, 2014)

  Subsequently, P-Brocard triangles have been mentioned in properties of points X_i for i = 110, 146, 147, 153, 599, and others.

  In October 2025, Peter Moses found barycentric coordinates for the P-Brocard triangle of a point P =\(p : q : r\), as follows:

  A-vertex = 2*a^2*p : (-b^2 + c^2)*p + a^2*(q + r) : (b^2 - c^2)*p + a^2*(q + r)

  B-vertex = (c^2 - a^2)*q + b^2*(r + p) : 2*b^2*q : (-c^2 + a^2)*q + b^2*(r + p)

  C-vertex = (-a^2 + b^2)*r + c^2*(p + q) : (a^2 - b^2)*r + c^2*(p + q) : 2*c^2*r

  The following triangles are perspective, with perspector X₃, for every point P:

  medial (TCCT 6.2)

  tangential (TCCT 6.5)

  1st circumperp (TCCT 6.21)

  2nd circumperp (TCCT 6.22)

  outer Napoleon (TCCT 6.31)

  inner Napoleon (TCCT 6.32)

  [and others].

  If P lies on the Stammler hyperbola (SH, the Feuerbach circumhyperbola of the tangential triangle), then the P-Brocard triangle is perspective to ABC, and the perspector lies on the cubic K028. The appearance of (i,j) in the following list means that the lies on SH, and the X_i-Brocard triangle is perspective to ABC with perspector X_j: (1,8), (3,3), (6,76), (155,847), (159,8743), (195,25043), (399,14254), (610,14256), [and others]

• X₇₀₂₇₀–X₇₀₂₇₇ — Points associated with the Warren H-circle

  based on notes from Benjamin Warren and Peter Moses, November, 2025. The Warren H-circle is introduced in the preamble just before X₆₇₂₁₂. Here the Warren H-triangle, A*B*C*, with vertices on the Warren H-circle, is introduced, and points on the H-circle are presented.

  In the plane of a triangle ABC, let

  A'B'C' = orthic triangle;

  A″ = reflection of H in line B'C', and define B″ and C″ cyclically;

  A_b = relection of A″ in line CA, and define B_c and C_a cyclically;

  A_c = relection of A″ in line AB, and define B_a and C_b cyclically;

  A* = midpoint of segment B_c C_b, and define B* and C* cyclically.

  The triangle A*B*C* is the Warren H-triangle, and A*,B*,C* lie on the Warren H-circle.

  Barycentrics \(x : y : z\) for vertex A* are given by

  x = a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10 + 3*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 3*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 2*b^6*c^4 - 3*a^4*c^6 - 6*a^2*b^2*c^6 - 2*b^4*c^6 + 3*a^2*c^8 + 3*b^2*c^8 - c^10

  y = b^2*(-a^2 + b^2 - c^2)*(-a^2 + b^2 + c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)

  z = -(c^2*(a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4))}

  The triangle A*B*C* is perspective to the following triangles:

  ABC (TCCT 6) at X₃

  ABC reflected in X₃ (TCCT 6.12) at X₃

  X₃ reflected in ABC (TCCT 6.13) at X₃

  circum-orthic (TCCT 6.20) at X₆₂₄₁

  Lucas central (MathWorld) at X₃

  [and others]

  The appearance of (i,j) in the following list means that X_i-of-A*B*C* = X_j: (1,185), (2,12022), (3,5), (4,44076), (5,45970), (8,18560), (9,50649), (10,13403), (11,6146), (20,14516), (21,54), (30,32423), (35,43831), (36,125), (40,12162), (56,67902), (80,32659), (100,4), (101,31850), (103,31848), (104,3), (105,18338),[and othersj]

  Warren H-circle = circumcircle-inverse of the line X₁₈₆ X₅₂₃, so that the circle passes through the circumcircle-inverse of X_i for these i: 186,523,14344,15412,19189,21445,23286,38294,39199,39201,39478,40631,40664,41204,42731,44375,44428,44680,44809,48383,48387,53255,53265,53304,57065,57089,57120,58756,66300,66512,68162,68741

  Warren H-circle = circumcircle-inverse of the line X₄₀₃ X₅₂₃, so that the circle passes through the circumcircle-inverse of X_i for these i: 403, 523, 6530, 14312, 14618, 15451, [and others].

  The Warren H-circle also passes through the points X_i for i = 70270..70277.

• X₇₁₁₈₃–X₇₁₃₁₅ — chi-points, X₇₁₁₈₃-X₇₁₂₃₅, and theta-points, (X₇₁₂₃₆-X₇₁₂₁₅

  contributed by Clark Kimberling and Peter Moses. January 20, 2026. Suppose P =\(p : q : r\) and U =\(u : v : w\) are points (barycentric coordinates), and define the chi-point of P and U, denoted by χ(P,U), as follows:

  χ(P,U) = p*(v^2 + w^2) : q*(w^2 + u^2) : r*(u:^2 + v^2)

  Geometrically,

  χ(P,U) = crosspoint[P, P / U^2]

  χ(P,U) = crosssum[gP, U^2 tP*X₆], where gP = isogonal conjugate of P, and tP = isotomic conjugate of P

  Define the theta-point of P and U, denoted by ϑ(P,U), as follows:

  ϑ(P,U) = u*(q*v + r*w) : v*(r*w + p*u) : w*(p*u + q*v)

  Geometrically, ϑ(P,U) = crosspoint of U and tP

  The appearance of (i,j),k in the following list means that χ(X_i, X_j) = k:

  (1,1),38; (1,2),1; (1,6),4118; (1,7),4319; (1,8),614; (1,9),21346;

  (2,1),141; (2,2),2; (2,4),6389; (2,6),626; (2,7),6554; (2,8),4000; (2,9),21258; (2,10),17045;

  (3,1),3917; (3,2),3; (3,3),13409; (3,4),426; (3,6),20819; (3,8),1473; (3,9),22440;

  (4,1),427; (4,2),4; (4,3),6747; (4,4),1899; (4,7),1863; (4,8),1851; (4,20),6619;

  (5,2),5;

  (6,1),39; (6,2),6; (6,4),39643; (6,6),20859; (6,7),30706; (6,8),16502; (6,9),23653;

  [and others].

  The appearance of (i,j),k in the next list means that ϑ(X_i, X_j) = k:

  (1,1),38; (1,2),10; (1,3),22069; (1,4),3914; (1,6),3778; (1,7),3663; (1,8),4847; (1,10),21020; (1,19),17872;

  (2,1),37; (2,2),2; (2,3),216; (2,4),6; (2,5),233; (2,6),39; (2,7),1; (2,8),9; (2,9),1212; (2,10),1213; (2,11),46101; (2,13),396; (2,14),395; (2,15),40695; (2,16),40696; (2,17),23302; (2,18),23303; (2,19),16583; (2,20),1249;

  (3,1),21318; (3,2),5; (3,3),13409; (3,4),427; (3,5),11197; (3,6),23635; (3,7),41007; (3,8),65684; (3,10),3136;

  (4,1),17441; (4,2),3; (4,3),3917; (4,4),1899; (4,6),6467; (4,7),41004;

  (5,1),21319; (5,2),140; (5,4),11245;

  (6,1),3721; (6,2),141; (6,3),22416; (6,4),5254; (6,6),20859; (6,7),3782; (6,8),40997; (6,10),21024; (6,13),53428; (6,14),53440; (6,17),53452; (6,18),53463; [and others].

  Note that χ(P,P) =ϑ(P,P) for all points P. Also,

  χ(X₁₂,X₁₀) = ϑ(X₈,X₁₂);

  χ(X₁₃,X₁) = ϑ(X₁₅,X₁₃);

  χ(X₁₄,X₁) = ϑ(X₁₆,X₁₄);

  χ(X₁₅,X₁) = ϑ(X₁₃,X₁₅);

  χ(X₁₆,X₁) = ϑ(X₁₄,X₁₆);

  As noted in Clark Kimberling and Peter Moses, “Line conjugates in the plane of a triangle” Aequationes mathematicae 29 (2023) 161-184, the mapping given by

  \(f(P,U) = p(v^{2} + w^{2}) - u (q v + r w)\): : = χ(P,U) - ϑ(P,U)

  is called the P-(B)line conjugate of U. The fact that this point lies on the line PU follows from the identity

  \(f(P,U) = s p - t u\): : , where s = u^2 + v^2 + w^2 and t = p*u + q*v + r*w.

  Geometrically, f(P,U) is the intersection of PU and the trilinear polar of the isotomic conjugate of U.

  For more, see the preamble just before X₇₁₃₁₆.

• X₇₁₃₁₆–X₇₁₃₄₃ — Differences and sums of χ-points and ϑ-points

  contributed by Peter Moses and Clark Kimberling. January 28, 2026. Let χ(P,U) and ϑ(P,U) be the mappings introduced in the preamble just before X₇₁₁₈₃; that is,

  χ(P,U) = p*(v^2 + w^2) : q*(w^2 + u^2) : r*(u:^2 + v^2)

  and

  ϑ(P,U) = u*(q*v + r*w) : v*(r*w + p*u) : w*(p*u + q*v) .

  Four kinds of triangle centers are represented in the range X₇₁₃₁₆-X₇₁₃₄₃:

  X₇₁₃₁₆ - X₇₁₃₂₁: χ(P,U) - χ(U,P)

  X₇₁₃₂₂ - X₇₁₃₂₈: χ(P,U) + χ(U,P)

  X₇₁₃₂₉ - X₇₁₃₃₇: ϑ(P,U) - ϑ(U,P)

  X₇₁₃₃₈ - X₇₁₃₄₃: ϑ(P,U) + ϑ(U,P),

  where P and U are selected from the set {X_i} for 1 <= i <= 20, and the four notations such as χ(P,U) - χ(U,P) refer to combos (as in the Introduction of ETC). That is, χ(P,U) and χ(P,U) are given by normalized barycentrics; otherwise, χ(P,U) and χ(P,U) would differ in degree so that - and + would not give triangle centers.

  Geometrically, χ(P,U) - χ(U,P) is the intersection of the line χ(P,U) χ(U,P) with the line at infinity, and χ(P,U) + χ(U,P) is the midpoint of χ(P,U) and χ(U,P).

  Barycentrics are as follows:

  χ(P,U) - χ(U,P) = q^3*u^3 + q^2*r*u^3 + q*r^2*u^3 + r^3*u^3 + q^2*r*u*v^2 + r^3*u*v^2 - p^3*v^3 - p*r^2*v^3 - p^3*v^2*w - p*q^2*v^2*w + q^3*u*w^2 + q*r^2*u*w^2 - p^3*v*w^2 - p*r^2*v*w^2 - p^3*w^3 - p*q^2*w^3 : :

  χ(P,U) + χ(U,P) = q^3*u^3 + q^2*r*u^3 + q*r^2*u^3 + r^3*u^3 + 2*p*q^2*u*v^2 + q^2*r*u*v^2 + 2*p*r^2*u*v^2 + r^3*u*v^2 + p^3*v^3 + p*r^2*v^3 + p^3*v^2*w + p*q^2*v^2*w + 2*p*q^2*u*w^2 + q^3*u*w^2 + 2*p*r^2*u*w^2 + q*r^2*u*w^2 + p^3*v*w^2 + p*r^2*v*w^2 + p^3*w^3 + p*q^2*w^3 : :

  Barycentrics for ϑ(P,U) - ϑ(U,P) and ϑ(P,U) + ϑ(U,P) are too long to be shown here.

  For i and j up to 100, ten points χ(X_i,X_j) - χ(X_j,X_i) are in Parts 1-35 of ETC; the appearance of (i,j,k) in the following list means that χ(X_i,X_j) - χ(X_j,X_i) = X_k:

  (1,2,5846), (1,23,9019), (1,75,740), (2,8,9053), (2,30, 30), (2,75,64870), (4,69,511), (6,76,732), (7,8,518), (63,92,8680)
  For i and j up to 100, two points χ(X_i,X_j) + χ(X_j,X_i) are in Parts 1-35 of ETC; the appearance of (i,j,k) in the following list means that χ(X_i,X_j) + χ(X_j,X_i) = X_k:

  (1,23,9019), (2,30,30)
  Points χ(X_i,X_j) - χ(X_i,X_j) in Part 36 of ETC are X₇₁₃₁₆-X₇₁₃₂₁; the appearance of (i,j,k) in the following list means that χ(X_i,X_j) - χ(X_j,X_i) = X_k:

  (2,3,71316), (2,5,71317), (2,6,71318), (2,7,71319), (2,10,71320), (2,20,71321)
  Points χ(X_i,X_j) + χ(X_j,X_i) in Part 36 of ETC are X₇₁₃₂₂-X₇₁₃₂₈; the appearance of (i,j,k) in the following list means that χ(X_i,X_j) + χ(X_j,X_i) = X_k:

  (1,2,71322), (2,3,71323), (2,4,71324), (2,6,71325), (2,7,71326), (2,8,71327), (2,10,71328)
  Points ϑ(X_i,X_j) - ϑ(X_j,X_i) in Part 36 of ETC are X₇₁₃₂₉-X₇₁₃₃₇; the appearance of (i,j,k) in the following list means that ϑ(X_i,X_j) - ϑ(X_j,X_i) = X_k:

  (1,7,71329), (1,10,71330), (2,5,71331), (2,9,71332), (2,10,71333), (2,17,71334), (2,18,71335), (4,6,71336), (4,7,71337)
  Points ϑ(X_i,X_j) + ϑ(X_j,X_i) in Part 36 of ETC are X₇₁₃₃₈-X₇₁₃₄₃; the appearance of (i,j,k) in the following list means that ϑ(X_i,X_j) + ϑ(X_j,X_i) = X_k:

  (1,7,71338), (1,8,71339), (1,10,71340), (2,5,71341), (3,4,71342), (7,9,71343)

• X₇₁₆₉₃–X₇₁₇₄₁ — Points related to the bicevian chordal triangle of X₁ and X₂

  contributed by Ivan Pavlov on March 5, 2026. Bicevian chordal triangles are defined in the preamble to X₆₀₆₁₄. This entry deals specifically with the bicevian chordal triangle of X₁ and X₂, henceforth in this preamble called TR.

  Some of it properties include:

  - TR is homothetic to the Aguilera-Pavlov triangle with center X₇₁₇₀₁.

  - The following tuples inidicate {triangle perspective to TR, perspector}: {ABC, X₃₀₅₇₁}, {excentral, X₅₂₁₅₅}, {incentral, X₃₇₉₅}, {medial, X₃₇₈₉}, {Gemini 3, X₂₈₆₀₀}, {Gemini 4, X₅₁₃}, {Gemini 7, X₁₀₀₂}

  The following orthology centers have also been established:

  X₁ - {Orthology center of 8th mixtilinear and TR}

  X₂ - {Orthology center of TR and intouch}

  X₁₆₅ - {Orthology center of 4th mixtilinear and TR}

  X₃₅₄ - {Orthology center of intouch and TR}

  X₅₉₀₂ - {Orthology center of reflections-of-X₁ and TR}

  X₆₀₉₂₈ - {Orthology center of Aguilera-Pavlov and TR}

  As usual, below, triangle names starting with CTR, refer to the catalog kept here.

• X₇₁₇₄₃–X₇₁₇₄₉ — Points associated with the Euler coaxal pencil of circles

  contributed by Peter Moses, March 12, 2026. Among the well known circles in the family of Euler coaxal pencil of circle are the following: circumcircle, nine-point circle, orthocentroidal circle, orthoptic of the Steiner inellipse, polar circle, and tangential circle. All members of the family are given by a parameter t and formulas for the square of the radius and center:

  radius squared = (-SA*SB*SC*(1 - 3*t)^2 + S^2*SW (-1 + t)^2) / (4*S^2) = R^2*(2 + t*(-3 + J^2*(-1 + 2*t))) / 2

  center = (-1 + t)*X₃ - t*X₄.

  equation of the circle: c^2*x*y + b^2*x*z + a^2*y*z - (x + y + z)*(SA*t*x + SB*t*y + SC*t*z) = 0

  Values of t for well-known circles are shown here:

  circumcircle: t = 0

  nine-point circle: t = 1/2

  orthocentroidal circle: t = 2/3

  orthoptic of the Steiner inellipse: t = 1/3

  polar circle: t = 1

  tangential circle: t = -((a^2b^2c^2)/(4 SA SB SC))
  The appearance of (n, {k(1), k(2), … k(m)}) in the following list means that the circle with center X_n and passes through the points X(k(i)) for i = 1..m: (6644, {3,25,5941,15551,35901,46408,46942,50363,50364,50381,53723,67356,67424,71745})

  (31861, {6,1344,1345,2453,11472,14685,14686,14687,15922,15928,70039,71746,71747})

  (11799, {3580,5523,5913,32111,47323,47324,47325})

  (7575, {187,1495,32110,47326,47327,56922,70569,71748})

  (7579, {1346,1347,6032,7699,7703,50718})

  (1012, {1,1709,47270,56411,70172,70207,71749})

  [and others].

  Suppose that P =\(p : q : r\) is any point, and let

  t = t(P) = (-2*(c^2*p*q + b^2*p*r + a^2*q*r))/((p + q + r)*((a^2 - b^2 - c^2)*p + (-a^2 + b^2 - c^2)*q + (-a^2 - b^2 + c^2)*r))

  Then there is a circle in the Euler pencil of circles that passes through P, given by the equation

  c^2*x*y + b^2*x*z + a^2*y*z - (x + y + z)*(SA*t*x + SB*t*y + SC*t*z) = 0,

  this being the circle with powers SA*t, SB*t, SC*t, and center on the Euler line given by

  a^2*(-a^2 + b^2 + c^2) + ((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - a^2*(-a^2 + b^2 + c^2))*t : :

  = 2*(-a^2 - b^2 + c^2)*(a^2 - b^2 + c^2)*(c^2*p*q + b^2*p*r + a^2*q*r) - a^2*(a^2 - b^2 - c^2)*((a^2 - b^2 - c^2)*p^2 - (a^2 - b^2 + c^2)*q^2 - (a^2 + b^2 - c^2)*r^2) : :

  and squared radius

  (-12*SA*SB*SC*t^2 + 2*(a^2*SA^2 + b^2*SB^2 + c^2*SC^2)*t^2 - a^2*b^2*c^2*(-2 + 3*t + J^2*t))/(8*S^2).

  The next list shows the parameters t for the foregoing list of circles.

  (6644, 2/(3 + J^2))

  (31861, (3*a^2*b^2*c^2)/(2*(a^2 + b^2 + c^2)*S^2))

  (11799, (3 + J^2)/(2*J^2))

  (7575, 3/(2*J^2))

  (7579, (2*(-3 + J^2))

  /(-9 + 4*J^2))

  (1012, R/(r + R))

  [and others].

  The above material about the Euler pencil of coaxal circles generalizes as follows:

  Suppose that two distinct circles have powers P{p1,p2,p3} and Q{q1,q2,q3}, so that equations for the circles are

  C(P) = c^2*x*y + b^2*x*z + a^2*y*z - (x + y + z)*(q1*x + q2*y + q3*z) = 0

  and

  C(Q) = c^2*x*y + b^2*x*z + a^2*y*z - (x + y + z)*(p1*x + p2*y + p3*z) = 0.

  Then the pencil of circles coaxal with these two are linear combinations of their equations:

  (1 - t)*C(P) + t*C(Q)

  Explicitly, the powers of the circles are

  (p1*(1 - t) + q1*t, p2*(1 - t) + q2*t, p3*(1 - t) + q3*t)

  and the radical axis is the line

  (p1 - q1)*x + (p2 - q2)*y + (p3 - q3)*z = 0.

Part 37

• X₇₂₃₅₃–X₇₂₃₉₁ — Points associated with equilateral limit curves

  based on notes contributed by Peter Moses, April, 22 2026. Many points, as functions of the sidelengths a,b,c of a triangle ABC with angles (and vertices) A, B, C have X₂ as a limiting point as A, B, C all approach π/3, or equivalently, as the sidelengths a,b,c all approach a common value. However, if a point P is a triangle center with first barycentric that approaches 0 as a,b,c approach a common value, it is possible that the limiting positions of P form a curve that we call an equilateral limit curve, the ELC of P. We allow this term to include the case that the limiting locus is, as a degenerate case, X₂. The ELC of X₁, X₂,…,X₁₀ is G, but the ELC of X₁₁ is the incircle, the ELC of X₁₄ is the circumcircle, and the ELC of X₅₉ is a deltoid.

  Points X₇₂₃₅₃-X₇₂₃₉₁ are triangle centers of the form

  P(k) = a^2 + b^2 + c^2 - b c - c a - a b + k (b - c)^2 : : ,

  where k is a real number. The ELC of P(k) is the circle

  (1 + k)*(3 + k)*(x + y + z)^2 - (3 + 2*k)^2*(x*y + x*z + y*z) == 0, which has center X₂ and radius k/(2k+3).

  Points in ETC before April 22, 2026 that have the form P(k) are represented by the following table, which lists each such point along with the associated value of k: X₅₂₈₈₅,-7 X₃₆₅₂₅, -9/2 X₉₀₃, -3 X₄₄₄₀, -2 X₄₄₀₉, -7/4 X₁₉₀, -1 X₃₆₅₂₂, -9/10 X₄₂₈₈₆, -7/9 X₄₃₇₀, -3/4 X₄₁₁₃₈, -3/5 X₄₄₂₂, -1/2 X₂, 0 X₄₀₄₈₀, 1/2 X₂₇₁₉₁, 1 X₁₀₈₆, infinity Points X₇₂₃₅₃-X₇₂₃₉₁ with associated values of k are shown next.

  X₇₂₃₅₃,-9 X(72354,-8) X(72355,-15/2) X₇₂₃₅₆,-6 X₇₂₃₅₇,-5 X₇₂₃₅₈,-4 X₇₂₃₅₉,-15/4 X₇₂₃₆₀,-7/2 X(72361,-10/3) X₇₂₃₆₂,-11/4 X₇₂₃₆₃,-8/3 X₇₂₃₆₄,-5/2 X₇₂₃₆₅,-7/3 X₇₂₃₆₆,-9/4 X₇₂₃₆₇,-5/3 X₇₂₃₆₈,-4/3 X₇₂₃₆₉,-5/4 X₇₂₃₇₀,-1/3 X₇₂₃₇₁,-1/4 X₇₂₃₇₂,1/4 X₇₂₃₇₃,1/3 X₇₂₃₇₄,2/3 X₇₂₃₇₅,3/4 X₇₂₃₇₆,5/4 X₇₂₃₇₇,4/3 X₇₂₃₇₈,3/2 X₇₂₃₇₉,2 X₇₂₃₈₀,9/4 X₇₂₃₈₁,5/2 X₇₂₃₈₂,3 X₇₂₃₈₃,7/2 X₇₂₃₈₄,4 X₇₂₃₈₅,9/2 X₇₂₃₈₆,5 X₇₂₃₈₇,6 X₇₂₃₈₈,7 X₇₂₃₈₉,15/2 X₇₂₃₉₀,8 X₇₂₃₉₁,9 For figures and equations involving circular ELCs, see Catalogue of points in ETC that have circular ELCs.

  For a figures and equations involving noncircular ELCs, see Catalogue of points in ETC that have noncircular ELCs.

• X₇₂₄₂₆–X₇₂₄₆₉ — Points related to a Miquel-like construction

  contributed by Ivan Pavlov, May 23, 2026. Let ABC be a triangle, P =\(u:v:w\) a point not on the BC, CA, or AB, and

  P_aP_bP_c = cevian triangle of P;

  O_a = circumcircle of PPbPc, and define O_b and O_c cyclically;

  A_c =O_a∩AB, and define B_a and C_b cyclically;

  A_b =O_a∩AC, and define B_c and C_a cyclically.

  The six points A_c, A_b, B_c, B_a, C_b, C_a lie on a conic, so the cross-triangle of B_cC_aA_b and C_bA_cB_a is degernerate as a line, L.

  Let L* = isotomic conjugate of the tripolar of L; then L* is given by the following barycentrics:

  u*(u+v)*(u+w)*(-a^2*v*w+c^2*v*(v+w)+b^2*w*(v+w)) : :

  The appearance of (i,j) in the following list means that if P = X_i then L* = X_j: (1,333), (2,599), (3,97), (4,2), (6,10130), (7,5231), (8,30827), (9,72426), (10,72427), (20,72428), (25,72429), (40,72430), (63,72431), (65,72432), (66,72433), (67,42008), (68,72434), (69,16051), (74,2986), (75,30984), (76,72435), [and others].

  Let P* = center of the conic of A_c, A_b, B_c, B_a, C_b, C_a. Then
  P* = a^2*v*(u+v)*w*(u+w) : :

  = isogonal conjugate of the complement of the isotomic conjugate of P.

  See X₇₂₄₃₉ - X₇₂₄₆₈.

• X₇₂₅₂₅–X₇₂₆₅₄ — Cross-concevian perspectors

  contributed by Ivan Pavlov on June 12th, 2026. Let T=ABC be a triangle and P'=(\(p:q:r\)) and P″=(\(u:v:w\)) two points. Let T'=A'B'C' be the P'-circumconcevian triangle of P″ and T″=A″B″C″ be the cevian triangle of P'.

  Define A* = AA″ n B'C', B* = BB”n C'A', C* = CC”nA’B'.

  The triangle A*B*C* is here named the {P’,P”}-cross-concevian triangle. It has A-vertex: (p (r u v + q u w + 2 p v w) : q r u v : q r u w) and is perspective to the cevian triangle of P″.

  The perspectoro is here named the {P’,P”}-cross-concevian perspector. It has barycentrics: p u (r v + q w) (r u v + q u w + 2 p v w) : :

  See Euclid 9598 for details.

• X₇₂₆₆₇–X₇₂₆₇₁ — Dao-Euler Points

  contributed by Clark Kimberling, July 13, 2026, based on notes from Dao Thanh Oai, César Lozada, and Peter Moses. From Dao Thanh Oai, June 16, 2026:

  In the plane of a triangle ABC, let

  P = “Kiepert point” with base angle α

  Q = isogonal conjugate of P;

  P' = Kiepert point with base angle -α

  Q' = isogonal conjugate of P'.

  Then PQ and P'Q' intersect on the Euler line.

  From César Lozada, June 16, 2026: Let K = PQ∩P'Q', and define f(α) = |OK|/|OH| = 1/(-1 + 4*cos^2α). Triangle centers associated with selected values of α appear in the following table. α K = K(α) f(α) 0 X₂ 1/3 π/12 X₁₄₈₁₃ (-1+\(\sqrt{3}\))/2 π/6 X₅ 1/2 π/4 X₄ 1 π/3 X₃₀ oo 5π/12 X₁₄₈₁₄ (-1-\(\sqrt{3}\))/2 π/2 X₂₀ -1 K(α) = 3a^4 - 2(b^2+c^2)a^2 - (b^2-c^2)^2 + 4a^2(-a^2+b^2+c^2) cos^2 α : :

  From Peter Moses, June 17, 2026:

  K(α) = 2*X₃ + sec(2α)*X₄ : :

  K(α) = S^2 + a^2 (a^2-b^2-c^2) sin^2 * α : :

  K(α) = 1 - (cot A)(cot B + cot C) sin^2 * α : :

  K(α) = 1 + (cot A)(cot A - cot ω)*sin^2 α : :

  K(α) = 3*X₂ + 4*sin^2(α)*X₃ (a combo)

  K(α) = (1 - 2 sin^2 α, 2 sin^2 α) (Shinagawa coefficients)
  Let t be the number such that (vector OP) = t*(vector OH).

  Then α = (1/2) arccos((1-t)/2t)) for t in (-oo,-1)∪[1/3,oo).

  Let t = (1 + \(\sqrt{5}\))/2, the golden ratio, and let t(n) = n*π/10 = -2 cos(n*π)/5). The next table shows triangle centers associated with selected values of n n; t(n) X_n 0 -2 X₇₂₆₆₇ 1 -t X₇₂₆₆₈ 2 1 - t X₇₂₆₆₉ 3 t - 1 X₇₂₆₇₀ 4 t X₇₂₆₇₁

• X₇₂₇₀₂–X₇₂₇₇₃ — Points related to parabicevian triangles

  contributed by Ivan Pavlov, June 28, 2026. The following definitions follow Euclid 9767, see the post for more details.

  Let Cev(X) denote the cevian triangle of some point X. Let U=(\(u:v:w\)) and P=(\(p:q:r\)), in barycentrics. Let T₁ be the paraidal triangle of Cev(P) wrt Cev(U) and let T₂ be the paraidal triangle of Cev(U) wrt Cev(P). Then we call the side triangle of T₁ and T₂ - parabicevian triangle of U and P.

  In this section we consider the proprties of some parabicevian triangles. The following are homothetic to ABC:

  the parabicevian of X₁ and X₇₅ - with center X₁₂₁₅ the parabicevian of X₃ and X₆₈ - with center X₁₈₄ the parabicevian of X₄ and X₆₉ - with center X₉₃₀₆ the parabicevian of X₇ and X₈ - with center X₁₃₇₆ the parabicevian of X₆ and X₇₆ - with center X₄₀₇₆ the parabicevian of X₁₀ and X₈₆ - with center X₇₂₇₆₉ In general, if Q is the isotomic conjugate of P, the parabicevian triangle of P and Q is homothetic to ABC with center at (v + w) (u^2 + v w) : : This is a sufficient, but not necessary condition.

  For arbitrary P and Q, the parabicevian triangle of P and Q is perspective to the medial triangle with center

  p u (r v - q w) (q r u (v - w) - p q (u + v) w + p r v (u + w)) : :

• X₇₂₈₁₈–X₇₂₈₃₇ — Points related to equiradial chain triangles

  contributed by Ivan Pavlov, July 27, 2026. Given a triangle ABC we construct three congruent circles c1, c2, c3, such that c1 is tangent to AB and BC, c3 is tangent to AC and BC, and c2 is tangent to c1, c3, and BC. Let O_a denote the center of c2, and define O_b and O_c cyclically. The triangle O_aO_bO_c is here named the 3C-equiradial chain triangle.

  A-vertex of OaObOC is the point (a^2 : a*b+2S : a*c+2S).

  Let T_a be the tangency point of c2 with BC, and define T_b and T_c cyclically. The triangle T_aT_bT_c is here named the 3T-equiradial chain triangle. The A-vertex of T_aT_bT_c is the point (0 : sc+2*r : sb+2*r). Both the 3C- and 3T-Equiradial chain triangles are perspective to ABC, with perspectors X₃₃₀₀ and X₇₂₈₃₂, respectivey. Note that X₇₂₈₃₂ lies on the rectangular circumhyperbola through X₃₃₀₀. The 3C-equiradial Chain triangle is also perspective to every CTR32 triangle.