X1016

X1-CROSS CONJUGATE OF X99

X1016 = isogonal conjugate of X1015
< 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) { 190, 514 }: Central Line L(190, 514) { 238, 519 }: Central Line L(238, 519) { 100, 667 }: Central Line L(100, 667) { 99, 813 }: Central Line L(99, 813) { 512, 660 }: Central Line L(512, 660) { 644, 666 }: Central Line L(644, 666) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_99 = STEINER POINT (0.1109, 0.4322) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_667 = CROSSDIFFERENCE OF X(2) AND X(37) (-6.1667, 5.7155) X_813 = o-(- 1, 1) CIRCUMCIRCLE POINT (0.9721, -0.0916) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_660 = TRILINEAR POLE OF LINE X(1)X(39) (0.9483, -0.0755) X_644 = TRILINEAR MULTIPLIER FOR YFF PARABOLA (-0.0171, 0.3359) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.0982, 0.1428) X_1016 = -CROSS CONJUGATE OF X(99) (0.3832, 0.2666) A B C X1 X99 X190 X514 X238 X519 X100 X667 X813 X512 X660 X644 X666 X1016 (a, b, c) = (5, 6, 7) | AB = 1

X1016 (X1-CROSS CONJUGATE OF X99) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/[\sin A \sin^{2}(A/2) [1 - \cos(B - C)]] : :\)
Barycentrics: \((b - c)^{- 2} : :\)
Search Value (6, 9, 13): \(2.541018928022\)
▸ Incident Central Lines7 lines

{8, 1083}, {99, 813}, {100, 667}, {190, 514}, {238, 519}, {512, 660}, {644, 666}

▸ Functional Relations7 relations

Notes

X1016 is the trilinear pole of line X100X190, which is the tangent to the circumcircle at X100 and to the Steiner circumellipse at X190. This line is also the locus of trilinear poles of tangents at P to hyperbola {{A, B, C,X2,P}}, as P moves on the Nagel line, and the locus of trilinear poles of tangents at P to hyperbola {{A, B, C,X6,P}}, as P moves on line X1X6. (Randy Hutson, October 15, 2018)

X1016 is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Nagel line. (Randy Hutson, October 15, 2018)

Let A5B5C5 and A6B6C6 be the Gemini triangles 5 and 6. Let A' be the barycentric product A5*A6 and define B' and C' cyclically. The lines AA', BB', CC' concur in X1016. (Randy Hutson, November 30, 2018)