X689

TRILINEAR POLE OF LINE X6X_76

X689 = Ψ(X32, X2)
< 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) { 111, 308 }: Central Line L(111, 308) { 110, 670 }: Central Line L(110, 670) { 6, 703 }: Central Line L(6, 703) { 1, 719 }: Central Line L(1, 719) { 2, 733 }: Central Line L(2, 733) { 82, 715 }: Central Line L(82, 715) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0103, -0.1475) X_111 = PARRY POINT (0.9885, -0.0457) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4915, 0.1700) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_670 = TRILINEAR POLE OF LINE X(2)X(39) (-0.0057, 0.2948) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_703 = EVEN (- 4, 0) CIRCUMCIRCLE POINT (1.0008, 0.2001) X_1 = INCENTER (0.5714, 0.2333) X_719 = EVEN (- 3, 1) CIRCUMCIRCLE POINT (1.0097, 0.1268) X_2 = CENTROID (0.5374, 0.2000) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (1.0071, 0.0450) X_82 = ISOGONAL CONJUGATE OF X(38) (0.5943, 0.2703) X_715 = EVEN (- 3, - 1) CIRCUMCIRCLE POINT (1.0000, 0.2040) X_689 = TRILINEAR POLE OF LINE X(6)X(76) (0.0585, 0.3579) A B C X688 X111 X308 X110 X670 X6 X703 X1 X719 X2 X733 X82 X715 X689 (a, b, c) = (5, 6, 7) | AB = 1

X689 (TRILINEAR POLE OF LINE X6X_76) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/[a^{3}(b^{4} - c^{4})] : :\)
Search Value (6, 9, 13): \(5.150634227494\)

X689 lies on the circumcircle.

▸ Incident Central Lines13 lines

{1, 719}, {2, 733}, {6, 703}, {75, 745}, {76, 755}, {82, 715}, {83, 729}, {110, 670}, {111, 308}, {251, 699}, {662, 787}, {741, 873}, {799, 813}

▸ Functional Relations6 relations

Notes

X689 is the center of the bianticevian conic of X1 and X75. This conic is a rectangular hyperbola passing through X1, X75, the excenters and their isotomic conjugates. (Randy Hutson, July 11, 2019)