X59–X97

Isogonal & Isotomic Conjugates of Classical Centers

Isogonal conjugates (X59–X74) and isotomic and cross conjugates (X75–X97) of the classical centers X1–X58.
< 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 nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) { 2, 3 }: Euler Line 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_5 = NINE-POINT CENTER (0.5561, 0.2489) X_2 = CENTROID (0.5374, 0.2000) X_59 = ISOGONAL CONJUGATE OF X(11) (0.5333, 0.4355) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5579, 1.5943) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5912, 1.0593) X_67 = ISOGONAL CONJUGATE OF X(23) (0.7606, -0.2152) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_70 = ISOGONAL CONJUGATE OF X(26) (0.7750, -1.1016) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5487, 0.1428) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_84 = ISOGONAL CONJUGATE OF X(40) (0.6071, 0.6816) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_90 = -CROSS CONJUGATE OF X(1) (0.1022, 0.0201) A B C X3 X1 X5 X2 X59 X64 X65 X66 X67 X68 X69 X70 X73 X74 X76 X80 X84 X88 X90 (a, b, c) = (5, 6, 7) | AB = 1

X59–X97 (Isogonal & Isotomic Conjugates of Classical Centers) — 15 representative centers (from 39 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_68 — PRASOLOV POINT

X68PRASOLOV POINT

Coordinates for X68 can be obtained easily from the Ceva ratios given his Prasolov’s proof of concurrence.