X861

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X9

X861 = crossdifference of every pair of points on line X57X647
< 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 { 650, 663 }: Central Line L(650, 663) 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_650 = CROSSDIFFERENCE OF X(1) AND X(3) (13.7143, -4.1991) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (8.6667, -3.2660) X_861 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(9) (-0.1634, -1.6336) A B C X2 X3 X4 X650 X663 X861 (a, b, c) = (5, 6, 7) | AB = 1

X861 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X9) 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 \(f(A,B,C) = \tan(C/2) \sin 2B \sin(C - A) + \tan(B/2) \sin 2C \sin(B - A)\)
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): \(4.916619474317\)
▸ Incident Central Lines2 lines

{2, 3}, {650, 663}

▸ Functional Relations1 relation

  • X861 = crossdifference of every pair of points on line X57X647

Notes

As a point on the Euler line, X861 has Shinagawa coefficients \((ES^{2}+2\$abS_{C}\$F-\$ab\$[2(E+F)F-S^{2}], -3ES^{2}+\$abS_{A}S_{B}\$-2\$ab\$S^{2})\).