X2883–X2966

Isogonal Conjugates with Respect to Special Triangles

Complementary (medial-isogonal), anticomplementary-isogonal, tangential-isogonal, and excentral-isogonal conjugates of classical triangle centers, together with trilinear and barycentric products.
< 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_2883 = COMPLEMENTARY CONJUGATE OF X(4) (0.5272, -0.4972) X_2884 = COMPLEMENTARY CONJUGATE OF X(7) (0.4949, 0.0541) X_2885 = COMPLEMENTARY CONJUGATE OF X(8) (0.6138, 0.2371) X_2886 = COMPLEMENTARY CONJUGATE OF X(9) (0.5316, 0.2039) X_2888 = ANTICOMPLEMENTARY CONJUGATE OF X(3) (0.4361, 0.1647) X_2889 = ANTICOMPLEMENTARY CONJUGATE OF X(5) (0.4036, 0.0095) X_2897 = ANTICOMPLEMENTARY CONJUGATE OF X(92) (0.3923, -0.1435) X_2898 = ORTHIC-ISOGONAL CONJUGATE OF X(7) (0.6478, 0.4473) X_2899 = ORTHIC-ISOGONAL CONJUGATE OF X(8) (1.2166, 0.5640) X_2900 = ORTHIC-ISOGONAL CONJUGATE OF X(9) (0.9198, 0.4906) X_2902 = ORTHIC-ISOGONAL CONJUGATE OF X(15) (0.7604, 0.4612) X_2903 = ORTHIC-ISOGONAL CONJUGATE OF X(16) (0.3913, 0.4165) X_2904 = ORTHIC-ISOGONAL CONJUGATE OF X(24) (0.5471, 0.6807) X_2914 = ORTHIC-ISOGONAL CONJUGATE OF X(186) (0.5079, 0.4982) X_2917 = TANGENTIAL-ISOGONAL CONJUGATE OF X(4) (0.4220, -0.4724) X_2924 = TANGENTIAL-ISOGONAL CONJUGATE OF X(14) (-0.0456, 0.6323) A B C X3 X1 X2883 X2884 X2885 X2886 X2888 X2889 X2897 X2898 X2899 X2900 X2902 X2903 X2904 X2914 X2917 X2924 (a, b, c) = (5, 6, 7) | AB = 1

X2883–X2966 (Isogonal Conjugates with Respect to Special Triangles) — 16 representative centers (from 84 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Isogonal Conjugates wrt Special Triangles (Isogonal Conjugates with respect to Special Triangles)

Suppose A'B'C' is the medial triangle of ABC and X is a triangle center wrt ABC. Let X' be the isogonal conjugate of X wrt A'B'C'. Then X' is a triangle center wrt ABC. In order to determine its center function g, the method of TCCT, p. 207 applies. Write X =\(x : y : z\) (wrt ABC). Then trilinears of X wrt A'B'C' are

(- ax + by + cz)/a : (ax - by + cz)/b : (ax + by - cz)/c,

so that the isogonal conjugate of X wrt A'B'C' is

a/(- ax + by + cz) : b/(ax - by + cz) : c/(ax + by - cz),

which, in trilinears wrt ABC, is \(g(a,b,c,x,y,z) : g(b,c,a,y,z,x) : g(c,a,b,z,x,y)\), where

\(g(a,b,c,x,y,z) = bc(ax - by - cz)[b^{2}(ax + by - cz) + c^{2}(ax - by + cz)]\)

This point is the isogonal conjugate of X wrt the medial triangle, or, more simply, the medial isogonal conjugate of X. In like manner, other conjugates of X wrt other triangles are defined. In this section, only isogonal conjugates are considered, and trilinears wrt to selected triangles are given by first trilinear wrt ABC, as follows:

MEDIAL ISOGONAL CONJUGATE OF X =\(x : y : z\)
bc(ax - by - cz)[\(b^{2}(ax + by - cz)\) + \(c^{2}(ax - by + cz)\)]
ANTICOMPLEMENTARY ISOGONAL CONJUGATE OF X =\(x : y : z\)
bc[- \(a^{2}\)/(by + cz) + \(b^{2}\)/(cz + ax) + \(c^{2}\)/(ax + by)]
ORTHIC-ISOGONAL CONJUGATE OF X =\(x : y : z\)
x(x cos A - y cos B - z cos C)
TANGENTIAL-ISOGONAL CONJUGATE OF X =\(x : y : z\)
a[- \(a^{3}\)/(cy + bz) + \(b^{3}\)/(az + cx) + \(c^{3}\)/(bx + ay)]

EXCENTRAL-ISOGONAL CONJUGATE OF X =\(x : y : z\)

  • a/[(b + c - a)(y + z)] + b/[(c + a - b)(z + x)] + c/[(a + b - c)(x + y)]

In each of the above cases,\(x : y : z\), as a triangle center, is given by trilinears \(h(a,b,c) : h(b,c,a) : h(c,a,b)\) for some function h. Accordingly, each of the first trilinears given above defines a triangle center \(f(a,b,c) : f(b,c,a) : f(c,a,b)\).

The term “complementary conjugate” already in use at the time of this writing (11/25/04) 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.

The subject of isogonal conjugation with respect to a central triangle continues at the preamble just before X6000.

X_2904 — ORTHIC-ISOGONAL CONJUGATE OF X_24

X2904ORTHIC-ISOGONAL CONJUGATE OF X24

Let P be an arbitrary point of the circumcircle , O, and let S(P) be the Simson-Wallace line of P, and P' the reflection of P in S(P). The locus of P' as P ranges through O is a bicircular circumquartic that meets the sidelines of ABC in 6 points other than A, B, C. The 6 points lie on a conic, here called the Montesdeoca conic, M. The center of M is X2904, and the perspector of M is X14111. Let u = \(S_{A}\), v = \(S_{B}\), w = \(S_{C}\), \(f(a,b,c) = u^{2}(S^{2} - v^{2})(S^{2} - w^{2})\), \(g(a,b,c) = 2vw(u^{2} - S^{2})(S^{2} + vw)\). A barycentric equation for M is
X_2963 — BARYCENTRIC PRODUCT X_17*X_18

X2963BARYCENTRIC PRODUCT X17*X18

,p> X2963 lies on these lines: 2,1225 5,2965 6,17 25,2934 37,2962 50,252 53,1487 93,393 111,930 141,2987 216,1989 230,251 566,2165 588,590 589,615 1879,2937