X50

X74-CEVA CONJUGATE OF X184

X50 = crosssum of X49 and X50
< 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) { 3, 6 }: Brocard Axis { 67, 248 }: Central Line L(67, 248) { 115, 231 }: Central Line L(115, 231) { 112, 477 }: Central Line L(112, 477) { 338, 401 }: Central Line L(338, 401) { 230, 858 }: Central Line L(230, 858) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_67 = ISOGONAL CONJUGATE OF X(23) (0.7606, -0.2152) X_248 = -LINE CONJUGATE OF X(132) (0.8229, -0.0790) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_231 = -CEVA CONJUGATE OF X(128) (1.0375, 0.6458) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_477 = TIXIER ANTIPODE (0.0176, -0.0644) X_338 = CEVAPOINT OF X(115) AND X(125) (0.6941, 0.0587) X_401 = BAILEY POINT (-0.0758, -1.4044) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_858 = COMPLEMENT OF X(23) (0.1126, -0.9116) X_50 = -CEVA CONJUGATE OF X(184) (2.1559, 2.8368) A B C X74 X184 X3 X6 X67 X248 X115 X231 X112 X477 X338 X401 X230 X858 X50 (a, b, c) = (5, 6, 7) | AB = 1

X50 (X74-CEVA CONJUGATE OF X184) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sin 3A : :\)
Barycentrics: \(\sin A \sin 3A : \sin B \sin 3B : \sin C \sin 3C\)
Tripolars: \((pending)\)
Search Value (6, 9, 13): \(3.400616763711\)
▸ Alternate Coordinates5 forms

  • Trilinears: \(\cos A \sin 2A + \sin A \cos 2A : :\)

  • Trilinears: \(\sin A + \cos A \cot D/2 : :\), where cot \(D/2 = (4 \operatorname{area})/(6R^{2} - a^{2} - b^{2} - c^{2})\), where R = abc/(4*area) (Peter Moses, 12/19/2011; cf. X568)

  • Trilinears: \(a(1 - 4 \cos^{2} A) : b(1 - 4 \cos^{2} B) : c(1 - 4 \cos^{2} C)\)

  • Trilinears: \(a(1 + 2 \cos 2A) : b(1 + 2 \cos 2B) : c(1 + 2 \cos 2C)\)

  • Barycentrics: \(a^{4} ((a^{2} - b^{2} - c^{2})^{2} - b^{2} c^{2}) : :\)

▸ Linear Combinations1 combination

  • X50 = \(-(r^{2} + 2rR + s^{2})(r^{2} + 4rR + 3R^{2} - s^{2})\) X1 + \(6rR(r^{2} + 4rR + 3R^{2} - s^{2})\) X2 + \(2r^{2}(r^{2} + 4rR + 3R^{2} - 3s^{2})\) X3 (Peter Moses, April 2, 2013)

▸ Incident Central Lines7 lines

{3, 6}, {67, 248}, {112, 477}, {115, 231}, {230, 858}, {338, 401}, {647, 654}

▸ Functional Relations18 relations

Notes

Let DEF be any equilateral triangle inscribed in the circumcircle of ABC. Let D' be the barycentric product E*F, and define E', F' cyclically. Then D',E',F' all line on a line passing through X50. In the special case that DEF is the circumtangential triangle, the points D',E',F' lie on the Brocard axis, and in case DEF is the circumnormal triangle, the points D',E'F' lie on the line X50X647. See also X6149. (Randy Hutson, January 29, 2015)

Let A'B'C' and A″B″C″ be the (equilateral) circumcevian triangles of X15 and X16. Let A* be the barycentric product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X50. See also X6149. (Randy Hutson, January 29, 2015)

Let AA1A2, BB1B2, CC1C2 be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the barycentric product A1*A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X50; see also X6149. (Randy Hutson, October 13, 2015)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the crossdifference of A1 and A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X50. (Randy Hutson, June 27, 2018)

X50 is the {X3,X6}-harmonic conjugate of X566.