X₂₉₆₇–X₃₀₀₀

MacBeath Points, Dual Centers DC(X) / CD(X) & Central Line Intersections

The eight MacBeath points (X₂₉₆₇–X₂₉₇₄), circle radical centers (X₂₉₇₅–X₂₉₇₈), dual triangle mappings DC(X) and CD(X) (X₂₉₇₉–X₂₉₉₈), the Danneels-Bevan homothetic center (X₂₉₉₉), and the Soddy-antiorthic intersection (X₃₀₀₀).

5, 6, 7 3, 4, 5 Kepler Kimberling

X₂₉₆₇–X₃₀₀₀ (MacBeath Points, Dual Centers DC(X) / CD(X) & Central Line Intersections) — 16 representative centers (from 34 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — MacBeath Points (Points lying on the MacBeath inconic)

The MacBeath inconic is the conic inscribed in triangle ABC having center X₅ and foci X₃ and X₄. An equation for this conic in barycentric coordinates is

\(u^{2}\)\(x^{2}\) + \(v^{2}\)\(y^{2}\) + \(w^{2}\)\(z^{2}\) - 2wvyz - 2wuzx - 2uvxy = 0,

where (u, v, w) = (sin 2A, sin 2B, sin 2C), i.e.,\(u : v : w\) = X₃.

In the parlance of TCCT, page 238, the MacBeath inconic is W(X₁₈₄), with trilinear equation

\(U^{2}\)\(x^{2}\) + \(V^{2}\)\(y^{2}\) + \(W^{2}\)\(z^{2}\) - 2WVyz - 2WUzx - 2UVxy = 0,

where \(U : V : W\) = X₁₈₄ = \(a^{2}\)cos A : \(b^{2}\)cos B : \(c^{2}\)cos C.

Points lying on the MacBeath inconic were contributed by Peter Moses (Hyacinthos, Nov. 12, 2004), and a selection of these are given here as X₂₉₆₇ to X₂₉₇₃. Comments written by Alexander Murray MacBeath are given in a Hyacinthos message of Dec. 12, 2004 from Antreas P. Hatzipolakis. See also MathWorld and more than a dozen Hyacinthos messages numbered between 10870 and 10923.

Macbeath authored articles entitled “The Deltoid I,” The Deltoid II,” and “The Deltoid III,” in Eureka. In the second of these, 11 (1949) 26-29, he introduces the conic as follows:

Lemma 1: Let O, H be the common points of a coaxal system of circles. Let a variable circle of the system cut the line of centres at C. Let T be a point on the circumference such that TC=k*OC, where k is a fixed ratio. Then the locus of T is a conic with foci at O,H. In Hyacinthos message 2009, dated Dec. 9, 2000, John Conway wrote This I know as “the MacBeath ellipse”, since Murray MacBeath wrote a nice article about it in “Eureka”, the Cambridge University undergraduate mathematical journal. Thus did the conic get its name. Points X₃₃₉, X₁₃₁₂, and X₁₃₁₃ lie on the MacBeath inconic. Peter Moses (Nov. 12, 2004) noted that if X is on this conic, then the reflection of X in X₅, and also the reflection of X in the Euler line, lie on the conic. He introduced the MacBeath circumconic, dual to the MacBeath inconic. The circumconic is given in barycentrics by

(sin 2A)yz + (sin 2B)zx + (sin 2C)xy = 0,

and in trilinears by

(cos A)yz + (cos B)zx + (cos C)xy = 0,

and it passes through X_i for I = 110, 287, 648, 651, 677, 895, 1331, 1332, 1797, 1813, 1814, 1815.

Group Preamble — Dual triangles, DC and CD Points (Dual triangles: DC points and CD points)

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. (Proof of similarity follows from Theorem 6E in TCCT, as the “gamma triangle” there is the dual of a triangle whose sidelines are respectively perpendicular to those of DEF.)

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'. The mapping DC is given for U =\(u : v : w\) (trilinears) by

DC(U) = vw/(bv + cw) : wu/(cw + au) : uv/(au + bv).

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.

If you have The Geometer’s Sketchpad, you can view DUAL TRIANGLE (DualTriangle.gsp).

The appearance of (i,j) in the following list means that X_j = DC(X_i):

1,81 2,6 3,275 4,2 5,288 6,83 7,1 8,57 9,1170 10,1171 20,1073 63,1172 64,801 66,76 67,671 68,2052 69,4 75,58 76,251 79,1255 80,88 81,2298 83,3108 85,1174 86,1126 92,284 95,1173 99,110 100,651 110,648 189,9 190,101 253,3 254,1993 255,829 256,1258 264,54 273,1167 280,222 286,943 290,98 309,947 311,1166 314,961 317,847 320,1168 321,1169 325,2065 329,282 330,31 335,1438 347,1433 485,588 486,589 513,1016 514,1252 519,2226 522,1262 523,249 596,593 648,112 651,1783 660,666 664,109 666,919 668,100 670,99 671,111 693,59 850,250 889,898 892,691 903,106 927,677 1000,89 1029,37 1031,39 1032,1249 1034,223 1121,2291 1131,493 1132,494 1138,323 1219,2221 1280,1462 1297,287 1305,1331 1441,1175 1494,74 1821,2311 1897,1461 1916,1976 1972,248 2113,239 2184,2287 2349,2341 2370,1797 2373,895 2481,105

The longest chain among the listed points is this one: 69 -→ 4 -→ 2 -→ 6 -→ 83 → 3108

Inversely, the circum-anticevian triangle of a point P is the dual of the cevian triangle of a point CD(P), given for P =\(p : q : r\) by the inverse of the DC-mapping; that is:

CD(P) = bc/(-a/p + b/q + c/r) : ca/(a/p - b/q + c/r) : ab/(a/p + b/q - c/r) = isogonal conjugate of X₆-Ceva conjugate of P.

The appearance of (i,j) in the following list means that X_j = CD(X_i):

1,7 2,4 3,253 4,69 6,2 9,189 31,330 37,1029 39,1031 54,264 57,8 58,75 59,693 74,1494, etc. (Just transpose the pairs in the list for DC(U).)

The vertices of the dual D'E'F' of the cevian triangle of a point U =\(u : v : w\) are given by

D' = 1/(b/w - c/v) : -1/(c/u + a/w) : 1/(a/v + b/u)

E' = 1/(b/w + c/v) : 1/(c/u - a/w) : -1/(a/v + b/u)

F' = -1/(b/w + c/v) : 1/(c/u + a/w) : 1/(a/v - b/u).

Triangle D'E'F' can be obtained from DEF by translation, dilation, and rotation. First translate DEF by vector O'O, where O' = circumcenter of D'E'F' and O = circumcenter of ABC. Then dilate D'E'F' so that its circumradius is that of ABC. Then rotate D'E'F' so that angle AOD’ is twice the angle between lines BC and EF.

Group Preamble — Intersections of Central Lines

This section presents points X₃₀₀₀ to X₃₀₁₉ of intersection of lines listed in MathWorld’s Central Line. Contributed by Eric Weisstein, December, 2004.

X₂₉₆₇1st MACBEATH POINT

X₂₉₆₇ = reflection of X₃₃₉ in X₅

X₂₉₆₈2nd MACBEATH POINT

X₂₉₆₈ = complement of X₁₈₉₇

X₂₉₆₉3rd MACBEATH POINT

X₂₉₆₉ = X₁₀₁₅-cross conjugate of X₁₀₈₆

X₂₉₇₀4th MACBEATH POINT

X₂₉₇₀ = X₈₄₇-Ceva conjugate of X₅₂₃

X₂₉₇₁5th MACBEATH POINT

X₂₉₇₁ = reflection of X₂₉₇₄ in X₅

X₂₉₇₂6th MACBEATH POINT

X₂₉₇₂ = crossdifference of every pair of points on line X₁₀₇X₁₁₂

X₂₉₇₃7th MACBEATH POINT

X₂₉₇₃ = cevapoint of X₁₉₈₆ and X₂₉₆₉

X₂₉₇₄8th MACBEATH POINT

X₂₉₇₄ = reflection of X₂₉₇₁ in X₅

X₂₉₇₅INSIMILICENTER(CIRCUMCIRCLE, AC-INCIRCLE)

X₂₉₇₅ is the insimilicenter(circumcircle, ac-incircle).

X₂₉₇₆RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, AC-INCIRCLE}

X₂₉₇₆ = anticomplement of X₂₅₀₅

X₂₉₇₇RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT CIRCLE, AC-INCIRCLE}

X₂₉₇₇ = center of circle {{X₁₀₀,X₁₂₀,X₅₅₂₁}}

X₂₉₇₈RADICAL CENTER OF {CIRCUMCIRCLE, BROCARD CIRCLE, CONWAY CIRCLE}

X₂₉₇₈ = X₇₈₅-Ceva conjugate of X₆

X₂₉₇₉CENTROID OF DUAL OF THE ORTHIC TRIANGLE

X₂₉₇₉ is the centroid of dual of the orthic triangle.

X₂₉₈₀ISOGONAL CONJUGATE OF X₂₉₇₉

X₂₉₈₀ is the isogonal conjugate of X₂₉₇₉.

X₂₉₈₁DC(X₁₃)

X₂₉₈₁ = perspector of ABC and unary cofactor triangle of outer Napoleon triangle

X₂₉₈₂DC(X₂₁)

X₂₉₈₂ = crosssum of X₁ and X₂₉₅₄

X₂₉₈₃DC(X₂₇)

X₂₉₈₃ = X₆₄₇-cross conjugate of X₁₀₁

X₂₉₈₄DC(X₅₃)

X₂₉₈₄ = X₂₄₈₅-cross conjugate of X₉₃₃

X₂₉₈₅DC(X₅₆)

X₂₉₈₅ = cevapoint of X₆ and X₈

X₂₉₈₆DC(X₇₄)

X₂₉₈₆ = trilinear pole of line X₃X₅₂₃ (the line of the Ehrmann cross-triangle)

X₂₉₈₇DC(X₉₈)

X₂₉₈₇ = cevapoint of X₆ and X₅₁₁

X₂₉₈₈DC(X₁₀₂)

X₂₉₈₈ = cevapoint of X₆ and X₅₁₅

X₂₉₈₉DC(X₁₀₃)

X₂₉₈₉ = X₁₉-isoconjugate of X₉₁₆

X₂₉₉₀DC(X₁₀₄)

X₂₉₉₀ = X₁₉-isoconjugate of X₉₁₂

X₂₉₉₁DC(X₁₀₅)

X₂₉₉₁ = reflection of X₁₃₃₂ in X₆

X₂₉₉₂CD(X₁₃)

X₂₉₉₂ = isogonal conjugate of X₃₁₂₉

X₂₉₉₃CD(X₁₄)

X₂₉₉₃ = isogonal conjugate of X₃₁₃₀

X₂₉₉₄CD(X₁₉)

X₂₉₉₄ = pole wrt polar circle of trilinear polar of X₁₀₆₈

X₂₉₉₅CD(X₂₁)

X₂₉₉₅ = trilinear pole of line X₉₀₅X₁₅₇₇

X₂₉₉₆CD(X₂₅)

X₂₉₉₆ = cevapoint of X₁₁₅ and X₅₂₅

X₂₉₉₇CD(X₂₈)

X₂₉₉₇ = cevapoint of X₁ and X₅₂₅

X₂₉₉₈CD(X₃₂)

X₂₉₉₈ = trilinear pole of line X₅₁₂X₆₂₅ (the complement of the Lemoine axis)

X₂₉₉₉DANNEELS-BEVAN HOMOTHETIC CENTER

Let \(R_{A}\) be the radical axis of the Bevan circle and the A-excircle. Define \(R_{B}\) and \(R_{C}\) cyclically. The three radical axes form a triangle homothetic to triangle ABC, and X₂₉₉₉ is the center of homothety. (Eric Danneels, Hyacinthos #10926, Dec. 5, 2004.

X₃₀₀₀(ANTIORTHIC AXIS)∩(SODDY LINE)

The antiorthic axis is line X₄₄X₅₁₃, more simply represented as L(1), meaning the line whose coefficients in trilinear coordinates are \(1 : 1 : 1\), these also being trilinears for X₁. The Soddy line is X₁X₇, alias L(657). For such identifications as these, use the MathWorld link just before X₃₀₀₀.