X3580

INTERSECTION OF LINES X2X_6 AND X30X_74

X3580 = complement of X323
< 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, 6 }: Centroid–Symmedian Line { 24, 68 }: Central Line L(24, 68) { 30, 74 }: Central Line L(30, 74) { 113, 403 }: Central Line L(113, 403) { 5, 568 }: Central Line L(5, 568) { 110, 468 }: Central Line L(110, 468) 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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_113 = JERABEK ANTIPODE (0.4319, 0.4718) X_403 = OF THE ORTHIC TRIANGLE (0.7040, 0.6358) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_568 = ORTHOHARMONIC OF X(50) (0.6437, 0.3393) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_468 = -LINE CONJUGATE OF X(3) (0.7498, 0.7558) X_3580 = INTERSECTION OF LINES X(2)X(6) AND X(30)X(74) (1.2481, 0.9637) A B C X2 X6 X24 X68 X30 X74 X113 X403 X5 X568 X110 X468 X3580 (a, b, c) = (5, 6, 7) | AB = 1

X3580 (INTERSECTION OF LINES X2X_6 AND X30X_74) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc[b^{6} + c^{6} + (a^{4} - b^{2}c^{2})(b^{2} + c^{2}) - 2a^{2}(b^{4} + c^{4}) + 2a^{2}b^{2}c^{2}]\)
Search Value (6, 9, 13): \(1.509706955694\)
▸ Alternate Coordinates1 form

  • Trilinears: \((\csc A)(1 + \cos 2B + \cos 2C) : :\)

X3580 lies on the Walsmith rectangular hyperbola.

▸ Incident Central Lines24 lines

{2, 6}, {5, 568}, {22, 1899}, {23, 1503}, {24, 68}, {30, 74}, {52, 1594}, {110, 468}, {113, 403}, {125, 511}, {140, 567}, {146, 1514}, {186, 2931}, {278, 2994}, {281, 445}, {287, 2373}, {297, 525}, {340, 687}, {451, 3193}, {467, 2052}, {542, 1495}, {1154, 2072}, {1352, 1995}, {1368, 2979}

▸ Functional Relations15 relations

Notes

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the orthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the orthic axis. The triangle A″B″C″ is homothetic to ABC, and its centroid is X3580; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)