X1005

INTERCEPT OF EULER LINE AND LINE X9X_100

X1005 = isogonal conjugate of X1242
< 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 { 9, 100 }: Central Line L(9, 100) { 55, 329 }: Central Line L(55, 329) { 108, 342 }: Central Line L(108, 342) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1242 = ISOGONAL CONJUGATE OF X(1005) (0.6074, 0.3216) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_9 = MITTENPUNKT (0.5110, 0.1615) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_329 = ISOTOMIC CONJUGATE OF X(189) (0.3847, -0.0222) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.5320, 0.6114) X_342 = ISOTOMIC CONJUGATE OF X(271) (0.5119, -0.1470) X_1005 = INTERCEPT OF EULER LINE AND LINE X(9)X(100) (0.5199, 0.1541) A B C X1242 X2 X3 X9 X100 X55 X329 X108 X342 X1005 (a, b, c) = (5, 6, 7) | AB = 1

X1005 (INTERCEPT OF EULER LINE AND LINE X9X_100) 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 \(f(a,b,c) = a^{5} - 2a^{4}(b + c) - a^{3}bc + a^{2}(2b^{3}, + 2c^{3} + b^{2}c + bc^{2}) - a(b^{4} + c^{4} - b^{3}c - bc^{3} - 4b^{2}c^{2})\), + bc(b - c)(\(b^{2}\) - \(c^{2})\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a\),b,c) As a point on the Euler line, X1005 has Shinagawa coefficients \(((E-2F)S^{2}+\$abS_{A}S_{B}\$+\$ab\$S^{2}, 2(E+F)S^{2}-2\$ab\$S^{2})\).
Search Value (6, 9, 13): \(3.734344464447\)
▸ Incident Central Lines4 lines

{2, 3}, {9, 100}, {55, 329}, {108, 342}

▸ Functional Relations1 relation