X249–X297

Isogonal Conjugates II & Hirst Inverses

Isogonal conjugates of centers X115–X248, together with cross conjugates and X2-Hirst inverses.
< 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_249 = ISOGONAL CONJUGATE OF X(115) (0.5045, 0.3948) X_250 = ISOGONAL CONJUGATE OF X(125) (0.5892, 0.5347) X_251 = ISOGONAL CONJUGATE OF X(141) (0.6177, 0.3042) X_252 = ISOGONAL CONJUGATE OF X(143) (0.4565, 0.1253) X_254 = -CROSS CONJUGATE OF X(4) (1.4506, 0.5587) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4217, 0.0365) X_258 = CONGRUENT INCIRCLES ISOSCELIZER POINT (0.5464, 0.1945) X_265 = REFLECTION OF X(3) IN X(125) (0.8606, -0.0500) X_268 = ISOGONAL CONJUGATE OF X(196) (0.7710, 1.5678) X_272 = ISOGONAL CONJUGATE OF X(209) (-0.2458, -0.0534) X_273 = ISOGONAL CONJUGATE OF X(212) (0.6191, 0.4329) X_280 = -CROSS CONJUGATE OF X(8) (0.6379, 0.8041) X_287 = -HIRST INVERSE OF X(98) (0.7437, -0.0511) X_290 = ISOGONAL CONJUGATE OF X(237) (0.7829, -0.1380) X_292 = -HIRST INVERSE OF X(291) (1.0127, 0.0417) X_297 = -HIRST INVERSE OF X(4) (0.8440, 1.0022) A B C X3 X1 X249 X250 X251 X252 X254 X255 X258 X265 X268 X272 X273 X280 X287 X290 X292 X297 (a, b, c) = (5, 6, 7) | AB = 1

X249–X297 (Isogonal Conjugates II & Hirst Inverses) — 16 representative centers (from 49 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Centers X249-X297 (isogonal conjugates of prevously listed centers)

Centers X249-X297 are isogonal conjugates of previously listed centers.

X_262 — ISOGONAL CONJUGATE OF X_182

X262ISOGONAL CONJUGATE OF X182

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X3 wrt Pa and define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X262. Also, X262 is also the isotomic conjugate, wrt A'B'C', of X3.