X864

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X32

X864 = crossdifference of every pair of points on line X76X647
< 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 { 669, 688 }: Central Line L(669, 688) 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_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-4.3514, 4.6163) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0103, -0.1475) X_864 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(32) (0.7834, 0.8435) A B C X2 X3 X4 X669 X688 X864 (a, b, c) = (5, 6, 7) | AB = 1

X864 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X32) 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 \(f(A,B,C) = \csc^{3} C \sin 2B \sin(C - A) + \csc^{3} B \sin 2C \sin(B - A)\)
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): \(-0.134897845191\)
▸ Incident Central Lines2 lines

{2, 3}, {669, 688}

▸ Functional Relations1 relation

  • X864 = crossdifference of every pair of points on line X76X647

Notes

As a point on the Euler line, X864 has Shinagawa coefficients \(((E + F)^{3}F - (E - 2F)FS^{2} + S^{4}, -(E + F)^{4} + 2(E^{2} - F^{2})S^{2} - S^{4})\).