X1070–X1114

Cross Conjugates, Trilinear Squares & Euler-Circumcircle Intersections

Cross conjugates, trilinear squares and cubes of classical centers, and the intersections of the Euler line with the circumcircle (X1113, X1114).
< 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_1070 = POINT ALKES (0.5823, 0.2764) X_1071 = INTERSECTION OF LINES X(1)X(84) AND X(4)X(7) (0.5633, 0.1308) X_1073 = -CROSS CONJUGATE OF X(3) (0.4792, -0.4343) X_1075 = -CEVA CONJUGATE OF X(4) (0.6630, 1.7036) X_1076 = POINT ALPHARD (0.3421, -0.3326) X_1078 = INTERSECTION OF LINES X(2)X(32) AND X(3)X(76) (0.4771, 0.1215) X_1079 = TRILINEAR SQUARE OF X(46) (0.6490, 0.5422) X_1086 = CENTER OF HYPERBOLA {{A,B,C,X(2),X(7)|} (0.7687, 0.1000) X_1090 = TRILINEAR SQUARE OF X(11) (0.8768, 0.0123) X_1092 = TRILINEAR CUBE OF X(3) (0.3503, 0.0118) X_1095 = TRILINEAR SQUARE OF X(16) (0.4127, 0.3906) X_1101 = TRILINEAR SQUARE OF X(110) (0.5344, 0.4306) X_1102 = TRILINEAR CUBE OF X(63) (0.2368, 0.0051) X_1106 = TRILINEAR SQUARE OF X(56) (0.6507, 0.4375) A B C X3 X1 X5 X2 X1070 X1071 X1073 X1075 X1076 X1078 X1079 X1086 X1090 X1092 X1095 X1101 X1102 X1106 (a, b, c) = (5, 6, 7) | AB = 1

X1070–X1114 (Cross Conjugates, Trilinear Squares & Euler-Circumcircle Intersections) — 14 representative centers (from 45 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_1081 — POINT ALPHECCA

X1081POINT ALPHECCA

Coordinates for X1081 are obtained from those of X554 by changing π/3 to - π/3; contributed by Edward Brisse.
X_1082 — POINT ALPHERATZ

X1082POINT ALPHERATZ

Coordinates for X1082 are obtained from those of X559 by changing π/3 to - π/3; contributed by Edward Brisse.
X_1088 — TRILINEAR SQUARE OF X_7

X1088TRILINEAR SQUARE OF X7

X1088 is the Brianchon point (perspector) of the inellipse that is the trilinear square of the Gergonne line. The center of this inellipse is X11019. (Randy Hutson, October 15, 2018)
X_1100 — COMPLEMENT OF X_319

X1100COMPLEMENT OF X319

X1100 is the midpoint of the bicentric pair \(y : z : x\) and \(z : x : y\), where \(x : y : z\) = X37
X_1111 — TRILINEAR SQUARE OF X_514

X1111TRILINEAR SQUARE OF X514

Let A7B7C7 and A8B8C8 be the Gemini triangles 7 and 8. Let A' be the trilinear product A7*A8 and define B' and C' cyclically. The lines AA', BB', CC' concur in X1111. (Randy Hutson, November 30, 2018)