X1008

INTERCEPT OF EULER LINE AND LINE X1X_76

X1008 = anticomplement of X37148
< 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 { 1, 76 }: Central Line L(1, 76) 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_1 = INCENTER (0.5714, 0.2333) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_1008 = INTERCEPT OF EULER LINE AND LINE X(1)X(76) (0.5404, 0.2077) A B C X2 X3 X4 X1 X76 X1008 (a, b, c) = (5, 6, 7) | AB = 1

X1008 (INTERCEPT OF EULER LINE AND LINE X1X_76) 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 \(f(a,b,c) = bc[a^{5}(b + c) + a^{4}(b + c)^{2} + a^{3}(b + c)(b^{2} + bc + c^{2})\), + \(a^{2}(b^{2} + c^{2} + bc)\)2 + abc(b + c)3 + \(b^{2}\)\(c^{2}(b + c)\)2]
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)
Search Value (6, 9, 13): \(2.493108164572\)
▸ Incident Central Lines2 lines

{1, 76}, {2, 3}

▸ Functional Relations1 relation

  • X1008 = anticomplement of X37148

Notes

As a point on the Euler line, X1008 has Shinagawa coefficients \(((E + F)^{3} + 2\$bc\$(E + F)^{2} - \$abS_{C}\$(E + F) + 2(E + F)S^{2}, \$bc\$S^{2})\).