X1217

CRETAN PERSPECTOR

Using barycentric coordinates, let P =\(p : q : r\) be a point not on a sideline of triangle ABC, and let P' be the isogonal conjugate of P. Let DEF and D'E'F' be the pedal triangles of P and P', respectively. Let

X = PD'∩P'D, Y = PE'∩P'E, Z = PF'∩P'F.

< 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) { 20, 97 }: Central Line L(20, 97) { 3, 393 }: Central Line L(3, 393) { 4, 394 }: Central Line L(4, 394) { 254, 378 }: Central Line L(254, 378) { 5, 1073 }: Central Line L(5, 1073) { 2, 1093 }: Central Line L(2, 1093) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1181 = 2nd SARAGOSSA POINT OF X(3) (0.5894, 0.1559) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_97 = ISOGONAL CONJUGATE OF X(53) (0.4798, 0.0756) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_393 = -CROSS CONJUGATE OF X(4) (0.6247, 0.5263) X_4 = ORTHOCENTER (0.6122, 0.3958) X_394 = -CEVA CONJUGATE OF X(3) (0.3778, 0.0284) X_254 = -CROSS CONJUGATE OF X(4) (1.4506, 0.5587) X_378 = REFLECTION OF X(22) IN X(3) (0.5857, 0.3262) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_1073 = -CROSS CONJUGATE OF X(3) (0.4792, -0.4343) X_2 = CENTROID (0.5374, 0.2000) X_1093 = TRILINEAR CUBE OF X(4) (0.6189, 0.5667) X_1217 = CRETAN PERSPECTOR (0.5642, 0.3204) A B C X1181 X20 X97 X3 X393 X4 X394 X254 X378 X5 X1073 X2 X1093 X1217 (a, b, c) = (5, 6, 7) | AB = 1

X1217 (CRETAN 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 \(f(A,B,C) = (\sec A)/(1 - \cos A \sin B \sin C)\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\tan A)/(1 - \cos A \sin B \sin C)\)
Search Value (6, 9, 13): \(22.939313972598\)
▸ Alternate Coordinates1 form

  • Barycentrics: \(h(a,b,c) : h(b,c,a) : h(c,a,b)\), where \(h(a,b,c) = 1/(a^{2} - 3b^{2} - 3c^{2})\) (Peter Moses, Oct. 22, 2012)

▸ Incident Central Lines6 lines

{2, 1093}, {3, 393}, {4, 394}, {5, 1073}, {20, 97}, {254, 378}

▸ Functional Relations2 relations

Notes

Let A'B'C' be the medial triangle of the orthic triangle of triangle ABC. Let A″ be the reflection of X4 in A', and define B″ and C″ cyclically. Let Kab and Kac be the circumcenters of triangles A″BA and A″CA, respectively. Let A’’’ = BKac∩CKab, and define B’’’ and C’’’ cyclically. The lines AA’’’, BB’’’, CC’’’ concur in X1217. (Antreas Hatzipolakis, Anopolis #39, 3/19/2002)

Then the triangle XYZ is perspective to ABC (Dominik Burek, June 8, 2012) and the perspector is given by barycentric coordinates \(k(a,b,c) : k(b,c,a) : k(c,a,b)\), where k = 1/(\(pqr(b^{2}\) + \(c^{2}\) - \(a^{2})\) + 2\(p(b^{2}\)\(r^{2}\) + \(c^{2}\)\(q^{2})\)) (Peter Moses, Oct. 22, 2012). The point X1217 results from taking P = X3.