X237–X248

Line Conjugates

Intersections of line PQ with the trilinear polar of the isogonal conjugate of P (P-line conjugates of Q).
< 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_237 = -LINE CONJUGATE OF X(2) (0.9894, 1.3825) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_240 = -LINE CONJUGATE OF X(48) (0.7457, 0.8594) X_241 = -LINE CONJUGATE OF X(55) (1.1217, 1.2442) X_242 = -LINE CONJUGATE OF X(71) (0.7507, 0.7161) X_243 = -LINE CONJUGATE OF X(73) (0.7516, 0.9505) X_244 = -LINE CONJUGATE OF X(100) (0.7857, 0.1166) X_245 = -LINE CONJUGATE OF X(110) (0.7358, 0.0716) X_246 = -LINE CONJUGATE OF X(110) (0.5857, -0.0517) X_247 = -LINE CONJUGATE OF X(110) (0.9546, 0.2514) X_248 = -LINE CONJUGATE OF X(132) (0.8229, -0.0790) A B C X3 X1 X237 X238 X240 X241 X242 X243 X244 X245 X246 X247 X248 (a, b, c) = (5, 6, 7) | AB = 1

X237–X248 (Line Conjugates) — 11 representative centers (from 12 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Centers X237-X248 (Line conjugates)

Centers X237-X248 are line conjugates. (The P-line conjugate of Q is the point where line PQ meets the trilinear polar of the isogonal conjugate of Q.)

See Clark Kimberling and Peter J. C. Moses, Line conjugates in the plane of a triangle, Aequationes Mathematicae, August 2022.

X_237 — X_3-LINE CONJUGATE OF X_2

X237-LINE CONJUGATE OF X2

X237 is the point of intersection of the Euler line and the Lemoine axis (defined as the radical axis of the circumcircle and the Brocard circle).