X269

ISOGONAL CONJUGATE OF X200

X269 is the isogonal conjugate of X200.
< 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 { 6, 57 }: Central Line L(6, 57) { 9, 241 }: Central Line L(9, 241) { 69, 200 }: Central Line L(69, 200) { 320, 326 }: Central Line L(320, 326) { 56, 738 }: Central Line L(56, 738) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_200 = -CEVA CONJUGATE OF X(9) (0.4392, 0.1037) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_9 = MITTENPUNKT (0.5110, 0.1615) X_241 = -LINE CONJUGATE OF X(55) (1.1217, 1.2442) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_320 = ISOTOMIC CONJUGATE OF X(80) (-2.6735, -3.5993) X_326 = ISOTOMIC CONJUGATE OF X(158) (0.3354, 0.0219) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_738 = ISOGONAL CONJUGATE OF X(728) (0.6449, 0.4466) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) A B C X200 X1 X7 X6 X57 X9 X241 X69 X320 X326 X56 X738 X269 (a, b, c) = (5, 6, 7) | AB = 1

X269 (ISOGONAL CONJUGATE OF X200) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\tan^{2} A/2 : \tan^{2} B/2 : \tan^{2} C/2\)
Barycentrics: \(\sin A \tan^{2} A/2 : :\)
Search Value (6, 9, 13): \(0.054966828794\)
▸ Alternate Coordinates4 forms

  • Trilinears: \([a^{2} - (b - c)^{2}]^{2} : :\)

  • Trilinears: \(1 - \sec^{2}(A/2) : :\)

  • Trilinears: \((1 - \cos A)/(1 + \cos A) : :\)

  • Trilinears: \((1 - \sec A)/(1 + \sec A) : :\)

▸ Incident Central Lines15 lines

{1, 7}, {3, 939}, {6, 57}, {9, 241}, {46, 1103}, {56, 738}, {69, 200}, {86, 1088}, {106, 934}, {142, 948}, {273, 1111}, {292, 1020}, {307, 936}, {320, 326}, {479, 614}

▸ Functional Relations10 relations

Notes

X269 is the vertex conjugate of the foci of the inellipse that is the trilinear square of the Gergonne line. The center of this inellipse is X11019, and the Brianchon point (perspector) is X1088. (Randy Hutson, October 15, 2018)

Let A1B1C1 be Gemini triangle 1. Let A' be the center of conic {A, B, C,B1,C1}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X269. (Randy Hutson, January 15, 2019)