X974

2nd EHRMANN POINT

X974 = midpoint of X125 and X185
< 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) { 125, 185 }: Central Line L(125, 185) { 6, 74 }: Central Line L(6, 74) { 5, 113 }: Central Line L(5, 113) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.6033, 0.1643) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_113 = JERABEK ANTIPODE (0.4319, 0.4718) X_974 = 2nd EHRMANN POINT (0.7025, 0.2393) A B C X125 X185 X6 X74 X5 X113 X974 (a, b, c) = (5, 6, 7) | AB = 1

X974 (2nd EHRMANN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \([sa^{10} - a^{8}(3b^{4} - 2b^{2}c^{2} + 3c^{4}) + a^{6}s(2b^{4} - 3b^{2}c^{2} + 2c^{4}) + a^{4}(b^{2} - c^{2})^{2}(2b^{4} - 7b^{2}c^{2} + 2c^{4}) - 3a^{2}s(b^{2} - c^{2})^{2}(b^{4} - 3b^{2}c^{2} + c^{4}) + (b^{2} - c^{2})^{4}(b^{4} + b^{2}c^{2} + c^{4})] \cos A\), where \(s = b^{2} + c^{2}\)
Search Value (6, 9, 13): \(7.602693635829\)
▸ Incident Central Lines2 lines

{5, 113}, {6, 74}

▸ Functional Relations4 relations

Notes

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

Let HaHbHc be the orthic triangle. Let La be the orthic axis of AHbHc, and define Lb and Lc cyclically. Let A″ = Lb∩Lc. B″ = Lc∩La, C″ = La∩Lb. Triangle A″B″C″ is inversely similar to ABC, with similitude center X6. The orthocenter of triangle A″B″C″ is X974. (Randy Hutson, January 29, 2018)

The pedal triangles of circumcenter and orthocenter with respect to orthic and medial triangles, respectively, are perspective, with perspector X974. (Angel Montesdeoca, August 29, 2019)