X2023

RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES

Let A'B'C' be the 1st Brocard triangle. Let A″ be the orthogonal projection of A on line B'C', and define B″ and C″ cyclically. Triangle A″B″C″ is perpsective to the medial triangle at X2023. (Note that the A-vertex of the medial triangle is the orthogonal projection of A' on line BC, and cyclically for B' and C'.) (Randy Hutson, July 11, 2019)
< 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) { 39, 115 }: Central Line L(39, 115) { 5, 39 }: Central Line L(5, 39) { 6, 98 }: Central Line L(6, 98) { 2, 694 }: Central Line L(2, 694) { 230, 511 }: Central Line L(230, 511) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_39 = BROCARD MIDPOINT (0.5778, 0.2305) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_98 = TARRY POINT (0.8891, -0.2281) X_2 = CENTROID (0.5374, 0.2000) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0179, 0.0378) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6642, 0.1572) A B C X39 X115 X5 X6 X98 X2 X694 X230 X511 X2023 (a, b, c) = (5, 6, 7) | AB = 1

X2023 (RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES) 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 \(f(A,B,C) = \sin \omega \cos(B - C) - \cos 2\omega \sin(A + \omega )\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.894584054186\)

X2023 lies on the half-Moses circle.

▸ Incident Central Lines10 lines

{2, 694}, {5, 39}, {6, 98}, {30, 2021}, {194, 1007}, {230, 511}, {325, 732}, {625, 736}, {1503, 2024}, {1575, 1738}

▸ Functional Relations4 relations