X1088

TRILINEAR SQUARE OF X7

X1088 is the Brianchon point (perspector) of the inellipse that is the trilinear square of the Gergonne line. The center of this inellipse is X11019. (Randy Hutson, October 15, 2018)
< 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) { 2, 85 }: Central Line L(2, 85) { 86, 269 }: Central Line L(86, 269) { 7, 354 }: Central Line L(7, 354) { 305, 341 }: Central Line L(305, 341) { 57, 658 }: Central Line L(57, 658) { 234, 555 }: Central Line L(234, 555) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1253 = ISOGONAL CONJUGATE OF X(1088) (0.5237, 0.1548) X_2 = CENTROID (0.5374, 0.2000) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5610, 0.2421) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5534, 0.2171) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_7 = GERGONNE POINT (0.5903, 0.2769) X_354 = WEILL POINT (0.5823, 0.2533) X_305 = ISOTOMIC CONJUGATE OF X(25) (0.3258, 0.0397) X_341 = ISOTOMIC CONJUGATE OF X(269) (0.3509, 0.0650) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_658 = TRILINEAR POLE OF LINE X(1)X(7) (0.3795, 0.5837) X_234 = -CEVA CONJUGATE OF X(177) (0.5946, 0.2875) X_555 = (X(1),X(158))-ANSWER TO QUESTION A (0.5982, 0.2990) X_1088 = TRILINEAR SQUARE OF X(7) (0.6052, 0.3210) A B C X1253 X2 X85 X86 X269 X7 X354 X305 X341 X57 X658 X234 X555 X1088 (a, b, c) = (5, 6, 7) | AB = 1

X1088 (TRILINEAR SQUARE OF X7) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sec^{4}(A/2) : :\)
Search Value (6, 9, 13): \(0.244629400364\)
▸ Incident Central Lines9 lines

{2, 85}, {7, 354}, {57, 658}, {75, 3668}, {86, 269}, {234, 555}, {272, 1014}, {305, 341}, {675, 934}

▸ Functional Relations4 relations

Notes

Let A'B'C' be the cross-triangle of the inner and outer Soddy triangles. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X1088. (Randy Hutson, December 10, 2016)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Conway triangles. Let A' be the trilinear product A1*A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X1088. (Randy Hutson, July 11, 2019)