X3234

VERTEX OF THE INSCRIBED YFF PARABOLA

Yff parabola animated over the 3-periodics in an elliptic billiard (i.e., the circumconic with center X9. Contributed by Dan Reznik, February 12, 2020.
< 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) { 101, 514 }: Central Line L(101, 514) { 118, 516 }: Central Line L(118, 516) { 1633, 2736 }: Central Line L(1633, 2736) { 100, 3239 }: Central Line L(100, 3239) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_118 = MIDPOINT OF X(4) AND X(101) (0.4133, 0.4603) X_516 = ISOGONAL CONJUGATE OF X(103) (0.7703, 0.7386) X_1633 = TCC-PERSPECTOR OF X(651) (0.2619, 0.5832) X_2736 = SR(X(103), X(105)) (0.9653, -0.1075) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_3239 = ISOGONAL CONJUGATE OF X(1461) (-7.7755, 1.1998) X_3234 = VERTEX OF THE INSCRIBED YFF PARABOLA (0.4055, 0.4423) A B C X101 X514 X118 X516 X1633 X2736 X100 X3239 X3234 (a, b, c) = (5, 6, 7) | AB = 1

X3234 (VERTEX OF THE INSCRIBED YFF PARABOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc(2a^{3} - b^{3} - c^{3} - a^{2}b - a^{2}c + b^{2}c + bc^{2})^{2}/(b - c)\)
Barycentrics: \((a - b) \cdot (a - c) \cdot (2 a^{3} - a^{2} b - b^{3} - a^{2} c + b^{2} c + b c^{2} - c^{3})^{2} : :\)
Search Value (6, 9, 13): \(1.029400045137\)

X3234 lies on the curve Q078, the Yff parabola.

▸ Incident Central Lines11 lines

{100, 3239}, {101, 514}, {118, 516}, {220, 31852}, {908, 24582}, {929, 28879}, {1145, 8074}, {1633, 2736}, {5845, 15634}, {9057, 32739}, {14116, 17044}

▸ Functional Relations8 relations

Notes

Properties of the inscribed Yff parabola:

  • center = X514

  • perspector = X190

  • focus = X101

  • directrix = X4X9, the trilinear pole of X1897

  • axis = X101X514

  • Simson line of focus = X118X516

  • pass-througth points: Xi for these i: 514, 649, 3234, 3239, 4024, 4375, 6544, 24979, 31182

  • points on axis: Xi for these i: 101, 514, 664, 666, 927, 3234, 3732, 4564, 14545, 14612, 34805, 34906, 36146

  • dual of Yff parabola: the circumconic {{A, B, C, X2, X7}}

  • square of latus recturm = ((b - c)^2*(c - a)^2*(a - b)^2*(a - b - c)^2*(a + b - c)^2*(a - b + c)^2*(a + b + c)^2)/(a^4 - a^3*b - a*b^3 + b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3 + c^4)^3

  • Contributed by Peter Moses, April 6, 2007 and February 4, 2010.