X862

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X19

X862 = crossdifference of every pair of points on line X63X647
< 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 { 661, 663 }: Central Line L(661, 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_661 = CROSSDIFFERENCE OF X(1) AND X(21) (48.4286, -23.0952) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (8.6667, -3.2660) X_862 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(19) (0.7262, 0.6939) A B C X2 X3 X4 X661 X663 X862 (a, b, c) = (5, 6, 7) | AB = 1

X862 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X19) 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 \(f(A,B,C) = \cot C \sin 2B \sin(C - A) + \cot B \sin 2C \sin(B - A)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(0.826096539919\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (a^{2} - bc)(b + c)/(b^{2} + c^{2} - a^{2})\) (M. Iliev, 5/13/07)

▸ Incident Central Lines2 lines

{2, 3}, {661, 663}

▸ Functional Relations1 relation

  • X862 = crossdifference of every pair of points on line X63X647

Notes

As a point on the Euler line, X862 has Shinagawa coefficients \((\$ab\$F, -\$abS_{C}\$)\).