X857

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X75

X857 = inverse-in-orthocentroidal-circle of X379
< 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 { 514, 661 }: Central Line L(514, 661) 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_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (48.4286, -23.0952) X_857 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(75) (-0.3133, -2.0259) A B C X2 X3 X4 X514 X661 X857 (a, b, c) = (5, 6, 7) | AB = 1

X857 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X75) 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 \(f(A,B,C) = \sin^{2} C \sin 2B \sin(C - A) + \sin^{2} B \sin 2C \sin(B - A)\)
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): \(8.299528356664\)
▸ Incident Central Lines2 lines

{2, 3}, {514, 661}

▸ Functional Relations5 relations

Notes

As a point on the Euler line, X857 has Shinagawa coefficients \((\$aS_{B}S_{C}\$, - \$a\$S^{2})\).