X866

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X513

X866 = crossdifference of every pair of points on line X100X647
< 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 { 244, 665 }: Central Line L(244, 665) 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_244 = -LINE CONJUGATE OF X(100) (0.7857, 0.1166) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.4579, 1.0988) X_866 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(513) (0.1513, -0.8103) A B C X2 X3 X4 X244 X665 X866 (a, b, c) = (5, 6, 7) | AB = 1

X866 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X513) 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 \(f(A,B,C) = \sin 2B \sin(A - C)/(\sin A - \sin B) - \sin 2C \sin(A - B)/(\sin A - \sin C)\)
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): \(5.564207379332\)
▸ Incident Central Lines2 lines

{2, 3}, {244, 665}

▸ Functional Relations1 relation

  • X866 = crossdifference of every pair of points on line X100X647

Notes

As a point on the Euler line, X866 has Shinagawa coefficients \(([(E+F)F+S^{2}]S^{2}+\$abS_{C}^{3}\$ -\$abS_{B}^{2}S_{C}\$+\$abS_{A}S_{B}\$(E+F) -\$abS_{C}\$[(E+F)^{2}-2S^{2}]-3\$ab\$FS^{2}, -[(E+F)^{2}+S^{2}]S^{2}-\$abS_{C}\$S^{2} +\$abS_{B}\$S^{2}+\$ab\$(E+F)S^{2})\).