X77

ISOGONAL CONJUGATE OF X33

X77 is the isogonal conjugate of X33.
< 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) { 1, 7 }: Soddy Line { 29, 34 }: Central Line L(29, 34) { 69, 73 }: Central Line L(69, 73) { 57, 81 }: Central Line L(57, 81) { 2, 189 }: Central Line L(2, 189) { 6, 241 }: Central Line L(6, 241) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_29 = CEVAPOINT OF INCENTER AND ORTHOCENTER (0.5903, 0.3384) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5487, 0.1428) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_2 = CENTROID (0.5374, 0.2000) X_189 = CYCLOCEVIAN CONJUGATE OF X(8) (0.6131, 0.7055) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_241 = -LINE CONJUGATE OF X(55) (1.1217, 1.2442) X_77 = ISOGONAL CONJUGATE OF X(33) (0.5270, 0.1305) A B C X33 X1 X7 X29 X34 X69 X73 X57 X81 X2 X189 X6 X241 X77 (a, b, c) = (5, 6, 7) | AB = 1

X77 (ISOGONAL CONJUGATE OF X33) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/(1 + \sec A) : 1/(1 + \sec B) : 1/(1 + \sec C)\)
Barycentrics: \(a/(1 + \sec A) : b/(1 + \sec B) : c/(1 + \sec C)\)
Search Value (6, 9, 13): \(-4.331307128236\)
▸ Alternate Coordinates4 forms

  • Trilinears: \(\cos A \sec^{2}(A/2) : \cos B \sec^{2}(B/2) : \cos C \sec^{2}(C/2)\)

  • Trilinears: \((b^{2} + c^{2} - a^{2})/(b + c - a) : :\)

  • Trilinears: \(S_{A}(S_{A} - bc) : :\)

  • Barycentrics: \(\cot A (1 - \cos A) : :\)

▸ Incident Central Lines19 lines

{1, 7}, {2, 189}, {6, 241}, {9, 651}, {29, 34}, {40, 947}, {55, 1037}, {56, 1036}, {57, 81}, {63, 219}, {65, 969}, {69, 73}, {75, 664}, {102, 934}, {283, 603}, {309, 318}, {738, 951}, {988, 1106}, {999, 1057}

▸ Functional Relations10 relations

Notes

In the plane of a triangle ABC, let

  • Ia = line through X7 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 X77 = radical center of Oa, Ob, Oc. See also X77. (Ivan Pavlov, April 1, 2022)