X3830

X3543com(EULER TRIANGLE)

X3830 = QA-P24 (AntiComplement of QA-P1 wrt the Morley Triangle) of quadrangle ABCX2 (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/52-qa-p24.html)
< 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 { 265, 541 }: Central Line L(265, 541) { 115, 1384 }: Central Line L(115, 1384) { 542, 1351 }: Central Line L(542, 1351) { 394, 1531 }: Central Line L(394, 1531) { 399, 1539 }: Central Line L(399, 1539) 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_265 = REFLECTION OF X(3) IN X(125) (0.8606, -0.0500) X_541 = DIRECTION OF VECTOR AX + BX + CX, where X = X(74) (0.7497, -0.3472) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_1384 = 2nd GRINBERG HOMOTHETIC CENTER (0.6750, 0.3910) X_542 = DIRECTION OF VECTOR AX + BX + CX, where X = X(98) (0.1649, 0.6534) X_1351 = REFLECTION OF X(3) IN X(6) (0.7000, 0.4324) X_394 = -CEVA CONJUGATE OF X(3) (0.3778, 0.0284) X_1531 = ORTHOJOIN OF X(32) (0.0445, -0.5419) X_399 = PARRY REFLECTION POINT (0.0032, 0.9936) X_1539 = ORTHOJOIN OF X(50) (0.4881, 0.6186) X_3830 = com(EULER TRIANGLE) (0.6497, 0.4936) A B C X2 X3 X4 X265 X541 X115 X1384 X542 X1351 X394 X1531 X399 X1539 X3830 (a, b, c) = (5, 6, 7) | AB = 1

X3830 (X3543com(EULER TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(5a^{4} - 4b^{4} - 4c^{4} - a^{2}b^{2} - a^{2}c^{2} + 8b^{2}c^{2} : :\)
Search Value (6, 9, 13): \(-9.829613512634\)
▸ Linear Combinations1 combination
▸ Incident Central Lines15 lines

{2, 3}, {115, 1384}, {265, 541}, {394, 1531}, {399, 1539}, {515, 3656}, {516, 3654}, {542, 1351}, {946, 3655}, {999, 3583}, {1159, 1836}, {1478, 3058}, {1853, 2777}, {3295, 3585}, {3521, 3527}

▸ Functional Relations11 relations

Notes

As a point on the Euler line, X3830 has Shinagawa coefficients \((1, -9)\).

Let T=A'B'C' be a triangle directly similar to ABC and inscribed in the circumcircle. As T varies, the isogonal conjugate of the trilinear product A'*B'*C' traces the Yff hyperbola, centered at X381 with vertices at X2 and X4, and foci at X3 and X3830. (Randy Hutson, June 27, 2018)