X1995

ORTHOCORRESPONDENT OF X6

Let P be a point in the plane of ABC and not on the circumcircle. Let (OA) be the circle tangent to the circumcircle at A and passing through P. Let A' be the antipode of A in (OA). Let LA be the tangent to (OA) at A'. Define LB and LC cyclically. Let TA = LB∩LC, and define TB and TC cyclically. Triangle TATBTC is homothetic to the tangential triangle. When P = X6, the center of homothety is X1995. (Randy Hutson, June 7, 2019)
< 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) circumcircle: Circumcircle of △ABC { 2, 3 }: Euler Line { 6, 110 }: Central Line L(6, 110) { 107, 264 }: Central Line L(107, 264) { 100, 344 }: Central Line L(100, 344) { 182, 373 }: Central Line L(182, 373) { 51, 576 }: Central Line L(51, 576) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_2 = CENTROID (0.5374, 0.2000) X_4 = ORTHOCENTER (0.6122, 0.3958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.5733, 0.6071) X_264 = ISOTOMIC CONJUGATE OF CIRCUMCENTER (0.5720, 0.3293) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_344 = ISOTOMIC CONJUGATE OF X(277) (0.6218, 0.2693) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X_373 = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID (0.5614, 0.2395) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_576 = ORTHOHARMONIC OF X(187) (0.6500, 0.3498) X_1995 = ORTHOCORRESPONDENT OF X(6) (0.5718, 0.2899) A B C X3 X2 X4 X6 X110 X107 X264 X100 X344 X182 X373 X51 X576 X1995 (a, b, c) = (5, 6, 7) | AB = 1

X1995 (ORTHOCORRESPONDENT OF X6) 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 \(f(a,b,c) = a(b^{4} + c^{4} - a^{4} - 4b^{2}c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.942234196243\)
▸ Incident Central Lines16 lines

{2, 3}, {6, 110}, {51, 576}, {98, 1302}, {100, 344}, {107, 264}, {157, 1624}, {169, 2000}, {182, 373}, {184, 575}, {197, 1621}, {251, 1184}, {323, 1351}, {1383, 1384}, {1611, 1627}, {1915, 2001}

▸ Functional Relations7 relations

Notes

As a point on the Euler line, X1995 has Shinagawa coefficients \((E - 2F, 2E + 2F)\).