X3509

INTERSECTION OF LINES X1X_32 AND X2X_7

X3509 = isogonal cojugate of X3512
< 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, 7 }: Centroid–Gergonne Line { 1, 32 }: Central Line L(1, 32) { 199, 228 }: Central Line L(199, 228) { 335, 385 }: Central Line L(335, 385) { 295, 511 }: Central Line L(295, 511) { 101, 758 }: Central Line L(101, 758) 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_7 = GERGONNE POINT (0.5903, 0.2769) X_1 = INCENTER (0.5714, 0.2333) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_199 = -CEVA CONJUGATE OF X(6) (0.4796, 0.0487) X_228 = -CEVA CONJUGATE OF X(71) (0.5231, 0.1126) X_335 = ISOTOMIC CONJUGATE OF X(239) (1.0387, 0.0318) X_385 = -HIRST INVERSE OF X(6) (2.2863, 2.0795) X_295 = ISOGONAL CONJUGATE OF X(242) (1.0582, 0.0142) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_758 = EVEN (1, 3) INFINITY POINT (0.1821, -0.2671) X_3509 = INTERSECTION OF LINES X(1)X(32) AND X(2)X(7) (5.8622, 7.9483) A B C X2 X7 X1 X32 X199 X228 X335 X385 X295 X511 X101 X758 X3509 (a, b, c) = (5, 6, 7) | AB = 1

X3509 (INTERSECTION OF LINES X1X_32 AND X2X_7) 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 \(f(a,b,c) = a^{3} - b^{3} - c^{3} + abc\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-21.044266084082\)

X3509 lies on the cubic Z(X385).

▸ Incident Central Lines20 lines

{1, 32}, {2, 7}, {6, 982}, {19, 2319}, {65, 2329}, {98, 813}, {101, 758}, {171, 846}, {199, 228}, {291, 1757}, {295, 511}, {335, 385}, {484, 1018}, {522, 649}, {529, 1146}, {594, 2160}, {666, 2311}, {902, 968}, {910, 1282}, {1054, 1575}

▸ Functional Relations4 relations