X859

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X81

X859 = midpoint of X851 and X855
< 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) { 851, 855 }: Euler Line { 2, 3 }: Euler Line { 56, 58 }: Central Line L(56, 58) { 36, 238 }: Central Line L(36, 238) { 198, 284 }: Central Line L(198, 284) { 81, 957 }: Central Line L(81, 957) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_851 = -HIRST INVERSE OF X(73) (1.4149, 2.4958) X_855 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(57) (0.8408, 0.9939) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_198 = -CEVA CONJUGATE OF X(6) (0.4741, -0.0381) X_284 = -CROSS CONJUGATE OF X(21) (0.5646, 0.2088) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_957 = ISOGONAL CONJUGATE OF X(956) (0.6245, 0.3599) X_859 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(81) (1.1279, 1.7448) A B C X851 X855 X2 X3 X56 X58 X36 X238 X198 X284 X81 X957 X859 (a, b, c) = (5, 6, 7) | AB = 1

X859 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X81) 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 \(f(A,B,C) = (\sin A + \sin B) \sin 2B \sin(C - A) + (\sin A + \sin C) \sin 2C \sin(B - A)\)
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): \(-1.990686716141\)
▸ Incident Central Lines7 lines

{2, 3}, {36, 238}, {56, 58}, {81, 957}, {198, 284}, {283, 945}, {333, 956}

▸ Functional Relations8 relations

Notes

As a point on the Euler line, X859 has Shinagawa coefficients \((\$aS_{A}\$ + \$a\$F,- \$aS_{A}\$ - \$a\$(E + F))\).