X3657

AYME PERSPECTOR

X3657 = isogonal conjugate of X3658
< 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, 513 }: Central Line L(3, 513) { 68, 521 }: Central Line L(68, 521) { 72, 523 }: Central Line L(72, 523) { 71, 661 }: Central Line L(71, 661) { 69, 693 }: Central Line L(69, 693) { 65, 924 }: Central Line L(65, 924) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3658 = ISOGONAL CONJUGATE OF AYME PERSPECTOR (0.8378, 0.9859) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_521 = ISOGONAL CONJUGATE OF X(108) (1.1224, 0.1534) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4367, 0.0733) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4807, 0.0915) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (48.4286, -23.0952) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_693 = ISOTOMIC CONJUGATE OF X(100) (-25.8163, 8.9981) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_924 = CROSSDIFFERENCE OF X(5) AND X(6) (0.2345, 0.7026) X_3657 = AYME PERSPECTOR (0.8264, -0.0756) A B C X3658 X3 X513 X68 X521 X72 X523 X71 X661 X69 X693 X65 X924 X3657 (a, b, c) = (5, 6, 7) | AB = 1

X3657 (AYME PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b^{2} - c^{2})/[2a(b + c)s_{B}s_{C} - S^{2} - S_{B}S_{C}]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(12.414237507689\)
▸ Incident Central Lines8 lines

{3, 513}, {65, 924}, {68, 521}, {69, 693}, {71, 661}, {72, 523}, {74, 915}, {895, 2990}

▸ Functional Relations1 relation

Notes

Let A'B'C' be the medial triangle of the reference triangle ABC. Let P be the point of intersection of the lines X1X3 and BC, and let P' be the point where the line through P perpendicular to line AX3 meets that line. Let LA be the line PP', and define LB and LC cyclically. Let A″=LB∩LC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X3657. (Jean-Louis Ayme, Hyacinthos #16676, August 21, 2008)

The Ayme triangle A″B″C″ is also perspective to these triangles: tangential, orthic, intangents, extangents, and the circumorthic. (Peter Moses, November 7, 2011)