X97

ISOGONAL CONJUGATE OF X53

X97 is the isogonal conjugate of X53.
< 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) { 3, 54 }: Central Line L(3, 54) { 2, 95 }: Central Line L(2, 95) { 216, 288 }: Central Line L(216, 288) { 110, 418 }: Central Line L(110, 418) { 276, 401 }: Central Line L(276, 401) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_54 = KOSNITA POINT (0.5880, 0.2176) X_2 = CENTROID (0.5374, 0.2000) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (0.5172, 0.1562) X_216 = -CEVA CONJUGATE OF X(51) (0.5201, 0.1353) X_288 = CEVAPOINT OF X(6) AND X(54) (0.5936, 0.2442) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_418 = -CEVA-CONJUGATE OF X(216) (0.4854, 0.0640) X_276 = ISOGONAL CONJUGATE OF X(217) (0.5554, 0.2771) X_401 = BAILEY POINT (-0.0758, -1.4044) X_97 = ISOGONAL CONJUGATE OF X(53) (0.4798, 0.0756) A B C X53 X3 X54 X2 X95 X216 X288 X110 X418 X276 X401 X97 (a, b, c) = (5, 6, 7) | AB = 1

X97 (ISOGONAL CONJUGATE OF X53) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cot A \sec(B - C) : \cot B \sec(C - A) : \cot C \sec(A - B)\)
Barycentrics: \(\cos A \sec(B - C) : \cos B \sec(C - A) : \cos C \sec(A - B)\)
Search Value (6, 9, 13): \(-13.339586550833\)
▸ Incident Central Lines5 lines

{2, 95}, {3, 54}, {110, 418}, {216, 288}, {276, 401}

▸ Functional Relations9 relations

Notes

In the plane of a triangle ABC, let

  • Ia = line through X3 parallel to BC, and define Ib and Ic cyclically.

  • Ac = Ia∩AB, and define Ba and Cb cyclically.

  • Ab = Ia∩AC, and define Bc and Ca cyclically.

  • Oa = circumcircle of A, Bc, Cb, and define Ob and Oc cyclically.

Then X97 = radical center of Oa, Ob, Oc. See also X77. (Ivan Pavlov, April 1, 2022)