X973

1st EHRMANN POINT

X973 = X442-of-orthic-triangle if ABC is acute
< 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) { 52, 1209 }: Central Line L(52, 1209) { 6, 24 }: Central Line L(6, 24) { 5, 51 }: Central Line L(5, 51) { 68, 568 }: Central Line L(68, 568) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_1209 = ISOGONAL CONJUGATE OF X(1166) (0.5121, 0.1912) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_568 = ORTHOHARMONIC OF X(50) (0.6437, 0.3393) X_973 = 1st EHRMANN POINT (0.6138, 0.3246) A B C X52 X1209 X6 X24 X5 X51 X68 X568 X973 (a, b, c) = (5, 6, 7) | AB = 1

X973 (1st EHRMANN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(u[a^{10} - 3a^{8}s + a^{6}(2b^{4} + 3b^{2}c^{2} + 2c^{4}) + a^{4}s(2b^{4} - b^{2}c^{2} + 2c^{4}) - a^{2}(b^{2} - c^{2})^{2}(3b^{4} + 5b^{2}c^{2} + s(b^{2} - c^{2})^{2}(b^{4} - b^{2}c^{2} + c^{4})] : :\), where \(u : v : w\) = X51 and \(s = b^{2} + c^{2}\)
Search Value (6, 9, 13): \(3.147032989910\)
▸ Incident Central Lines3 lines

{5, 51}, {6, 24}, {68, 568}

▸ Functional Relations3 relations

Notes

For constructions of X973 and X974, see Hyacinthos message 3695, Sept. 1, 2001, and related messages.

Let HaHbHc be the orthic triangle. X973 is the radical center of the nine-point circles of AHbHc, BHcHa, CHaHb. (Randy Hutson, January 29, 2018)