X3613

ISOTOMIC CONJUGATE OF X1078

X3613 is the isotomic conjugate of X1078.
< 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) { 5, 141 }: Central Line L(5, 141) { 4, 160 }: Central Line L(4, 160) { 53, 232 }: Central Line L(53, 232) { 311, 325 }: Central Line L(311, 325) { 66, 2548 }: Central Line L(66, 2548) { 184, 2980 }: Central Line L(184, 2980) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1078 = INTERSECTION OF LINES X(2)X(32) AND X(3)X(76) (0.4771, 0.1215) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_4 = ORTHOCENTER (0.6122, 0.3958) X_160 = -OF-TANGENTIAL-TRIANGLE (0.4841, -0.0647) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_232 = -CEVA CONJUGATE OF X(132) (0.6919, 0.7779) X_311 = ISOTOMIC CONJUGATE OF X(54) (0.4818, 0.1803) X_325 = -HIRST INVERSE OF X(69) (-0.3370, -0.7398) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5912, 1.0593) X_2548 = INSIMILICENTER(MOSES CIRCLE, (X(4),2R)) (0.5840, 0.2606) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_2980 = ISOGONAL CONJUGATE OF X(2979) (0.6001, 0.7109) X_3613 = ISOTOMIC CONJUGATE OF X(1078) (0.5844, 0.2955) A B C X1078 X5 X141 X4 X160 X53 X232 X311 X325 X66 X2548 X184 X2980 X3613 (a, b, c) = (5, 6, 7) | AB = 1

X3613 (ISOTOMIC CONJUGATE OF X1078) 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 \(f(a,b,c) = bc/(b^{2}c^{2} + c^{2}a^{2} + a^{2}b^{2} - a^{4})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-5.990491369595\)
▸ Linear Combinations1 combination
▸ Incident Central Lines9 lines

{4, 160}, {5, 141}, {53, 232}, {66, 2548}, {157, 3425}, {184, 2980}, {311, 325}, {1316, 3447}, {1485, 3148}

▸ Functional Relations4 relations

Notes

Let A' be the point of intersection of the tangents to the nine-point circle at the points where the circle meets line BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X3613. Also, A'B'C' is the side-triangle of the tangential triangle of the medial triangle and the tangential triangle of the orthic triangle. (Randy Hutson, August 30, 2011.)

X3613 is the pole of the Lemoine axis with respect to the nine-point circle. (Luis González, Hyacinthos #20253, October 5, 2011)

X3613 is the perspector of the nine-point circle and lies on the hyperbola that passes through the points A, B, C, X4, X5. (Randy Hutson, December 30, 2012.)

Let LA be the radical axis of the nine-point circle and the circle having AN as diameter, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector is X3613. See Vijay Krishna, X3613.pdf (X(3613).pdf), (November 13, 2019)