X3233

VERTEX OF THE INSCRIBED KIEPERT PARABOLA

X3233 = intercept of pedal and antipedal lines of X110
< 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) { 30, 113 }: Central Line L(30, 113) { 114, 468 }: Central Line L(114, 468) { 110, 476 }: Central Line L(110, 476) { 99, 1304 }: Central Line L(99, 1304) { 1302, 1576 }: Central Line L(1302, 1576) { 355, 3109 }: Central Line L(355, 3109) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_113 = JERABEK ANTIPODE (0.4319, 0.4718) X_114 = KIEPERT ANTIPODE (0.3616, 0.4140) X_468 = -LINE CONJUGATE OF X(3) (0.7498, 0.7558) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_476 = TIXIER POINT (0.9824, 0.2685) X_99 = STEINER POINT (0.1109, 0.4322) X_1304 = COLLINGS TRANSFORM OF X(403) (0.7822, 0.5273) X_1302 = COLLINGS TRANSFORM OF X(381) (0.3827, 0.5987) X_1576 = ISOGONAL CONJUGATE OF X(850) (0.3750, 0.6166) X_355 = FUHRMANN CENTER (0.5408, 0.2645) X_3109 = ORTHOGONAL PROJECTION OF X(1) ON EULER LINE (0.5529, 0.2404) X_3233 = VERTEX OF THE INSCRIBED KIEPERT PARABOLA (0.4343, 0.4780) A B C X30 X113 X114 X468 X110 X476 X99 X1304 X1302 X1576 X355 X3109 X3233 (a, b, c) = (5, 6, 7) | AB = 1

X3233 (VERTEX OF THE INSCRIBED KIEPERT PARABOLA) 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 \(f(a,b,c) = bc(a^{2} - b^{2})(a^{2} - c^{2})[-2a^{4} + (b^{2} + c^{2})a^{2} + (b^{2} - c^{2})^{2}]^{2}\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.674042487553\)
▸ Incident Central Lines6 lines

{30, 113}, {99, 1304}, {110, 476}, {114, 468}, {355, 3109}, {1302, 1576}

▸ Functional Relations1 relation

  • X3233 = intercept of pedal and antipedal lines of X110

Notes

Contributed by Fred Lang, April 1, 2007.