X3602–X3662

Morley Cubics, Medial Homothetic Centers H(X; M, t) & Kirikami-Schiffler Points

Centers associated with the three Morley cubics K029–K031 (X3602–X3615), medial-triangle t-dilation homothetic centers H(X; M, t) and their isogonal conjugates H⁻¹(X; M, t) (X3616–X3637), Soddy-Gergonne, Kirikami, Alkaid, and Almach points (X3638–X3646), and Kirikami-Schiffler points KS(T) (X3647–X3662).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_3602 = ISOGONAL CONJUGATE OF X(3274) (0.5856, 0.2621) X_3606 = ISOGONAL CONJUGATE OF X(3276) (0.5988, 0.8316) X_3612 = INTERSECTION OF LINES X(1)X(3) AND X(21)X(90) (0.5439, 0.1827) X_3620 = H(X(6); M, -2) (0.4873, 0.1462) X_3621 = H(X(8); M, 2) (0.2653, -0.0667) X_3625 = H-1(X(1); M, -1/2) (0.4184, 0.0833) X_3627 = H-1(X(3); M,-1/2) (0.6684, 0.5426) X_3629 = H-1(X(6); M, 1/2) (0.6939, 0.3681) X_3630 = H-1(X(6); M, 1/2) (0.3184, -0.0354) X_3632 = H-1(X(8); M, 1/2) (0.3673, 0.0333) X_3633 = H-1(X(8); M, -1/2) (0.7755, 0.4332) X_3648 = KS(ANTICOMPLEMENTARYTRIANGLE) (0.4170, -0.0039) X_3657 = AYME PERSPECTOR (0.8264, -0.0756) X_3658 = ISOGONAL CONJUGATE OF AYME PERSPECTOR (0.8378, 0.9859) X_3659 = FEUERBACH POINT OF EXCENTRAL TRIANGLE (0.1962, 0.5121) X_3660 = MIDPOINT OF {X(3513), X(3514)} (0.8653, 0.7732) A B C X3 X1 X3602 X3606 X3612 X3620 X3621 X3625 X3627 X3629 X3630 X3632 X3633 X3648 X3657 X3658 X3659 X3660 (a, b, c) = (5, 6, 7) | AB = 1

X3602–X3662 (Morley Cubics, Medial Homothetic Centers H(X; M, t) & Kirikami-Schiffler Points) — 16 representative centers (from 61 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Points Associated with Morley Cubics (Points associated with Morley cubics K29, K30, K31)

Points X3602-X3609 are associated with the three Morley Cubics indexed as K29, K30, K31 at Bernard Gibert’s Cubics in the Triangle Plane. The points are related to those in the section just before X3272, under the heading “Points Associated with Equilateral Triangles.” See also

Jean-Pierre Ehrmann and Bernard Gibert, “A Morley Configuration,” :cite:`Forum Geometricorum <forum-geometricorum>` 1 (2001) 51-58. (Click here to download a PDF of this article.)

Group Preamble — Homothetic Centers: X3616-X3637

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), as summarized here:

TABLE 1: H(X; M, t) X t=1/2 t=2 t=-1/2 t=-2 X1 X3616 X145 X667 X3617 X3 X631 X20 X3090 X3091 X4 X3091 X3146 X3523 X3522 X5 X1656 X4 X3526 X631 X6 X3618 X193 X3619 X3620 X8 X3617 X3621 X3622 X3623 X10 X1698 X8 X3624 X3616

TABLE 2: \(H^{-1}(X; M, t)\) X t=1/2 t=2 t=-1/2 t=-2 X1 X3244 X1125 X3625 X3626 X3 X550 X140 X3627 X546 X4 X382 X5 X1657 X550 X5 X546 X3628 X548 X3530 X6 X3630 X3589 X3631 X3632 X8 X3633 X10 X3634 X3244 X10 X3626 X3635 X3636 X3637

Group Preamble — Kirikami-Schiffler Points: X3647-X3652

Suppose that X is a point and A'B'C' is a central triangle. Let LA be the line through A' parallel to the Euler line of triangle BCX, let LB be the line through B' parallel to the Euler line of CXA, and let LC 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 LA, LB, LC 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. A summary follows:

triangle A'B'C' LA∩LB∩LC ABC X79 medial X3647 excentral X191 anticomplementary X3648 intouch X3649 extouch X3650 Feuerbach X442 Fuhrmann X191 1st circumperp X3651 2nd circumperp X21 Carnot X3652

Suppose that A'B'C' is the cevian triangle of a point P. Then LA, LB, LC concur if and only if P lies on the cubic K455.

Suppose that A'B'C' is the anticevian triangle of a point P. Then LA, LB, LC concur if and only if P lies on the cubic given by the barycentric equation

a(2a + b + c)yz(y - z) + b(2b + c + a)zx(z - x) + c(2c + a + b)xy(x - y) = 0.

Suppose that A'B'C' is the pedal triangle of a point P on the line X1X3. Then LA, LB, LC concur.

Suppose that A'B'C' is the antipedal triangle of a point P on the line X1X4. Then LA, LB, LC concur.

X_3608 — 1st MORLEY-GIBERT PERSPECTOR

X36081st MORLEY-GIBERT PERSPECTOR

X3608 is the perspector of the Morley triangle and the triangle of the cusps of the Steiner deltoid. (Bernard Gibert, November 3, 2010). See
X_3609 — 2nd MORLEY-GIBERT PERSPECTOR

X36092nd MORLEY-GIBERT PERSPECTOR

X3609 is the perspector of the triangle of the cusps of the Steiner deltoid and the equilateral triangle J defined just before X3272. (Bernard Gibert, November 3, 2010)
X_3610 — 1st AYME-MOSES PERSPECTOR

X36101st AYME-MOSES PERSPECTOR

Moses found that the locus of X such that the cevian triangle of X is perspective to the Ayme triangle is a cubic which passes through the points Xi for i = 1, 2, 19, 75, 279, 304, 346, 2184. A barycentric equation for this Ayme-Moses cubic follows:

(Cyclic sum of ayz[\(by(a^{2}\) + \(b^{2}\) - \(c^{2})\) - \(cz(a^{2}\) - \(b^{2}\) + \(c^{2}\)] ) = 0.

X_3611 — 2nd AYME-MOSES PERSPECTOR

X36112nd AYME-MOSES PERSPECTOR

In a Hyacinthos message dated January 7, 2011, Jean-Louis Ayme noted that the orthic triangle of ABC is perspective to the medial triangle of the extangents triangle of ABC. Peter Moses found coordinates for the perspector, X3611.
X_3612 — INTERSECTION OF LINES X_1X_3 AND X_21X_90

X3612INTERSECTION OF LINES X1X_3 AND X21X_90

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the reflection triangle of X1 at X3612. (Randy Hutson, July 31 2018)
X_3614 — 1st HUTSON-FEUERBACH POINT

X36141st HUTSON-FEUERBACH POINT

Let EuaEubEuc be the Feuerbach triangle, L the line through A and X5, and A″ = L∩EubEuc; define B″ and C″ cyclically. Then the lines EuaA″, EubB″, EucC″ concur in X3614. X3614 = {X5,X12}-harmonic conjugate of X11. Further, X3614 is the trilinear pole (with respect to the Feuerbach triangle) of the perspectrix of ABC and the Feuerbach triangle. (Randy Hutson, August 30, 2011.)
X_3616 — H(X_1; M, 1/2)

X3616H(X1; M, 1/2)

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. Let LA be the external tangent to circles OB and OC that is the nearer of the two to OA. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Then A'B'C' is homothetic to ABC, and the center of homothety is X3616. See the reference at X1001.
X_3624 — H(X_10; M, 1/2)

X3624H(X10; M, 1/2)

X3624 = homothetic center of ABC and cross-triangle of Aquila and anti-Aquila triangles
X_3628 — H⁻¹(X_5; M, 2)

X3628H-1(X5; M, 2)

For another construction see Antreas Hatzipolakis and Peter Moses, Euclid 2576.
X_3632 — H⁻¹(X_8; M, 1/2)

X3632H-1(X8; M, 1/2)

X3632 = homothetic center of outer Garcia triangle and mid-triangle of medial and anticomplementary triangles
X_3635 — H⁻¹(X_10; M, -1/2)

X3635H-1(X10; M, -1/2)

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X3635 and X15519 are equal.
X_3638 — INNER SODDY-GERGONNE POINT

X3638INNER SODDY-GERGONNE POINT

Write I = X1, U = X7, r = inradius, R = circumradius, and d = 4*\(\sqrt{3}\)*area(ABC). Then distance ratios are given by |IP|/|PU| = 2r(r + 4R)/d and |IQ|/|QU| = -2r(r + 4R)/d
X_3640 — 1st KIRIKAMI POINT

X36401st KIRIKAMI POINT

j(a,b,c) = \(a^{4}\) + \(b^{4}\) + \(c^{4}\) - \(8b^{3}\)c - \(8bc^{3}\) + \(6b^{2}\)\(c^{2}\) - \(2a^{2}\)\(b^{2}\) - \(2a^{2}\)\(c^{2}\) + 8abc(b+c-a).
X_3645 — POINT ALKAID

X3645POINT ALKAID

Denote the incenter and excenters by I, IA, IB, IC. Let JA be the incenter of triangle BCI, and define JB and JC cyclically. The lines IAJA, IBJB, ICJC concur in X3645.
X_3646 — POINT ALMACH

X3646POINT ALMACH

Denote the incenter and excenters by I, IA, IB, IC. Let KA be the centroid of triangle BCI, and define KB and KC cyclically. The lines IAKA, IBKB, ICKC concur in X3646.
X_3649 — KS(INTOUCH TRIANGLE)

X3649KS(INTOUCH TRIANGLE)

Two proofs of Dao’s theorem about cyclic hexagons were published in 2014: Nikolaos Dergiades’s Dao’s Theorem on Six Circumcenters Associated with a Cyclic Hexagon and Telv Cohl’s A Purely Synthetic Proof of Dao’s Theorem on Six Circumcenters Associated with a Cyclic Hexagon. In the degenerate case that the hexagon is the intouch triangle A'B'C', the triangle of the circumcenters of AB'C', A'BC', A'B'C is perspective to A'B'C' with perspector X3649. (Dao Thanh Oai, Francisco Javier Garcia Capitan, ADGEOM #1718, September 16, 2014)
X_3651 — KS(1st CIRCUMPERP TRIANGLE)

X3651KS(1st CIRCUMPERP TRIANGLE)

For a construction of X3651 as the orthocenter of a certain triangle, see: Antreas Hatzipolakis and César Lozada, Hyacinthos 29288.
X_3652 — KS(CARNOT TRIANGLE)

X3652KS(CARNOT TRIANGLE)

Quim Castellsaguer defines the Carnot triangle as the triangle formed by the circumcenters of the triangles BCH, CAH, ABH, where H denotes the orthocenter. The Carnot triangle is also known as the Johnson triangle.