X1009

INTERCEPT OF EULER LINE AND LINE X1X_39

X1009 = crossdifference of every pair of points on line X647X659
< 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, 39 }: Central Line L(1, 39) { 72, 672 }: Central Line L(72, 672) { 518, 583 }: Central Line L(518, 583) { 283, 1065 }: Central Line L(283, 1065) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1244 = ISOGONAL CONJUGATE OF X(1009) (0.5903, 0.2249) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_39 = BROCARD MIDPOINT (0.5778, 0.2305) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4367, 0.0733) X_672 = CROSSDIFFERENCE OF X(1) AND X(514) (1.9245, 2.2184) X_518 = ISOGONAL CONJUGATE OF X(105) (0.9154, 0.6489) X_583 = ORTHOGONAL OF X(500) (0.6638, 0.3726) X_283 = -CROSS CONJUGATE OF X(21) (0.4475, 0.0704) X_1065 = ISOGONAL CONJUGATE OF X(1064) (0.5777, 0.2607) X_1009 = INTERCEPT OF EULER LINE AND LINE X(1)X(39) (0.5532, 0.2412) A B C X1244 X2 X3 X1 X39 X72 X672 X518 X583 X283 X1065 X1009 (a, b, c) = (5, 6, 7) | AB = 1

X1009 (INTERCEPT OF EULER LINE AND LINE X1X_39) 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 \(f(a,b,c) = a^{4}(b + c) + 2a^{3}bc - a^{2}(b^{3} + c^{3}) + bc(b + c)(b^{2} + c^{2})\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)
Search Value (6, 9, 13): \(1.996211929923\)
▸ Incident Central Lines5 lines

{1, 39}, {2, 3}, {72, 672}, {283, 1065}, {518, 583}

▸ Functional Relations2 relations

Notes

As a point on the Euler line, X1009 has Shinagawa coefficients \((2\$bc\$(E + F)^{2} + \$bcS_{B}S_{C}\$ - 2\$bc(S_{A})^{2}\$ - 4\$bc\$S^{2}, \$bc\$S^{2})\).