X842

ISOGONAL CONJUGATE OF X542

X842 is the antipode of X691 on the circumcircle.
< 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) { 691, 3 }: Central Line L(691, 3) { 30, 99 }: Central Line L(30, 99) { 23, 110 }: Central Line L(23, 110) { 112, 186 }: Central Line L(112, 186) { 2, 476 }: Central Line L(2, 476) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_691 = TRILINEAR POLE OF LINE X(6)X(110) (1.0101, 0.0886) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_99 = STEINER POINT (0.1109, 0.4322) X_23 = FAR-OUT POINT (1.3871, 2.4232) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.7957, 0.8758) X_2 = CENTROID (0.5374, 0.2000) X_476 = TIXIER POINT (0.9824, 0.2685) X_842 = ISOGONAL CONJUGATE OF X(542) (-0.0101, 0.1156) A B C X691 X3 X30 X99 X23 X110 X112 X186 X2 X476 X842 (a, b, c) = (5, 6, 7) | AB = 1

X842 (ISOGONAL CONJUGATE OF X542) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a/(2ax - by - cz) : :\), where \(x = x(A,B,C) = \sec(A + \omega )\)
Barycentrics: \(a^{2}/[2a^{6} - 2a^{4}(b^{2} + c^{2}) + a^{2}(b^{4} + c^{4}) - (b^{2} - c^{2})^{2}(b^{2} + c^{2})] : :\)
Search Value (6, 9, 13): \(14.192926745701\)
▸ Incident Central Lines11 lines

{2, 476}, {3, 691}, {4, 935}, {23, 110}, {30, 99}, {74, 512}, {98, 523}, {107, 468}, {111, 647}, {112, 186}, {858, 925}

▸ Functional Relations21 relations

  • X842 = reflection of X691 in X3

  • X842 = isogonal conjugate of X542

  • X842 = trilinear pole of line X6X526

  • X842 = Λ(X6, X13)

  • X842 = Ψ(X6, X526)

  • X842 = intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,X4,X23}}

  • X842 = reflection of X98 in the Euler line

  • X842 = reflection of X74 in the Brocard axis

  • X842 = inverse-in-circle-O(15,16) of X112

  • X842 = inverse-in-{circumcircle, nine-point circle}-inverter of X3258

  • X842 = inverse-in-Moses-radical-circle of X111

  • X842 = SR(P,U), where P and U are the circumcircle intercepts of the Fermat axis

  • X842 = 2nd-Parry-to-ABC similarity image of X23

  • X842 = 3rd-Parry-to-circumsymmedial similarity image of X110

  • X842 = X10787-of-McCay-triangle

  • X842 = X2709-of-circumsymmedial-triangle

  • X842 = Cundy-Parry Phi transform of X14246

  • X842 = Cundy-Parry Psi transform of X14357

  • X842 = trilinear pole, wrt Thomson triangle, of line X110X5085

  • X842 = trilinear pole, wrt circumcevian triangle of X511, of Brocard axis

  • X842 = orthoptic-circle-of-Steiner-circumellipse-inverse of X14731

Notes

Let L = X74X98 or any line parallel to X74X98. Let La be the line of reflection of L in line BC, and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X842. (Randy Hutson, February 10, 2016)

Let A'B'C' be the Artzt triangle. Let A″ be the reflection of A' in BC, and define B″ and C″ cyclically. Let A″’ be the reflection of A in B'C', and define B″’ and C″’ cyclically. Let A* = B″B″’∩C″C″’, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X842. (Randy Hutson, December 10, 2016)