X867

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X514

X867 = crossdifference of every pair of points on line X101X647
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 3 }: Euler Line { 11, 244 }: Central Line L(11, 244) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_244 = -LINE CONJUGATE OF X(100) (0.7857, 0.1166) X_867 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(514) (0.3379, -0.3222) A B C X2 X3 X4 X11 X244 X867 (a, b, c) = (5, 6, 7) | AB = 1

X867 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X514) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sin C \sin 2B \sin(A - C)/(\sin A - \sin B) - \sin B \sin 2C \sin(A - B)/(\sin A - \sin C)\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A)f(A,B,C)\)
Search Value (6, 9, 13): \(4.084846711476\)
▸ Incident Central Lines2 lines

{2, 3}, {11, 244}

▸ Functional Relations1 relation

  • X867 = crossdifference of every pair of points on line X101X647

Notes

As a point on the Euler line, X867 has Shinagawa coefficients \(((E-2F)S^{2}+\$abS_{C}^{2}\$+\$abS_{A}S_{B}\$ -\$abS_{C}\$(E+F), -2(E+F)S^{2}+\$ab\$S^{2})\).