X863

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X31

X863 = crossdifference of every pair of points on line X75X647
< 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 { 667, 788 }: Central Line L(667, 788) 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_667 = CROSSDIFFERENCE OF X(2) AND X(37) (-6.1667, 5.7155) X_788 = ODD (- 3, 0) INFINITY POINT (1.0251, -0.1264) X_863 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(31) (0.8506, 1.0193) A B C X2 X3 X4 X667 X788 X863 (a, b, c) = (5, 6, 7) | AB = 1

X863 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X31) 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 \(f(A,B,C) = \csc^{2} C \sin 2B \sin(C - A) + \csc^{2} B \sin 2C \sin(B - A)\)
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): \(-0.431852977608\)
▸ Incident Central Lines2 lines

{2, 3}, {667, 788}

▸ Functional Relations1 relation

  • X863 = crossdifference of every pair of points on line X75X647

Notes

As a point on the Euler line, X863 has Shinagawa coefficients \((\$abS_{A}S_{B}\$+\$abS_{C}\$F, -\$abS_{C}^{2}\$ -\$ab\$S^{2})\).