X952

INTERSECTION OF X1X_5 AND X3X_8

As the isogonal conjugate of a point on the circumcircle, X952 lies on the line at infinity.
< 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, 5 }: Feuerbach Line { 3, 8 }: Central Line L(3, 8) { 4, 145 }: Central Line L(4, 145) { 10, 140 }: Central Line L(10, 140) { 30, 511 }: Central Line L(30, 511) { 40, 550 }: Central Line L(40, 550) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0000, 0.2041) X_1 = INCENTER (0.5714, 0.2333) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_8 = NAGEL POINT (0.4694, 0.1333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_40 = BEVAN POINT (0.4286, -0.0292) X_550 = MIDPOINT OF X(3) AND X(20) (0.4439, -0.0448) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.1267, 0.6191) A B C X953 X1 X5 X3 X8 X4 X145 X10 X140 X30 X511 X40 X550 X952 (a, b, c) = (5, 6, 7) | AB = 1

X952 (INTERSECTION OF X1X_5 AND X3X_8) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc[2a^{4} - 2a^{3}(b + c) - a^{2}(b^{2} - 4bc + c^{2}) + (2a - b - c)(b - c)(b^{2} - c^{2})] : :\)
▸ Incident Central Lines12 lines

{1, 5}, {3, 8}, {4, 145}, {10, 140}, {30, 511}, {40, 550}, {150, 664}, {182, 996}, {390, 1000}, {546, 946}, {547, 551}, {572, 594}

▸ Functional Relations7 relations

  • X952 = isogonal conjugate of X953

  • X952 = crossdifference of every pair of points on line X6X654

  • X952 = X30-of-Fuhrmann-triangle

  • X952 = inner-Garcia-isogonal conjugate of X40

  • X952 = X5663-of-excentral-triangle

  • X952 = (inverse-in-incircle)-isogonal conjugate of X33645

  • X952 = Cundy-Parry Psi transform of X14260

Notes

Let Ia be the reflection of X1 in the perpendicular bisector of BC, and define Ib and Ic cyclically. X952 = X517 of IaIbIc, which is the inner Garcia triangle.. (Randy Hutson, September 14, 2016)

Let A'B'C' be the outer Garcia triangle and A″B″C″ the inner Garcia triangle. Let A* be the isogonal conjugate, wrt A″B″C″, of A', and define B* and C* cyclically. The lines A″A*, B″B*, C″C* concur in X952; see also X355. (Randy Hutson, December 2, 2017)