X1133

BURGESS POINT

Rotate line BC about B away from A through angle B/3, and rotate line BC about C away from A through angle C/3; let A' be the point in which the two rotated lines meet. Define B' and C' cyclically. Let A″ be the point of intersection of lines BC' and B'C, and define B″ and C″ cyclically. The lines AA', BB', CC' concur in X357, and AA″, BB″, CC″ concur in X358. The first of these with reference to triangle A'B'C' is X1133; i.e., X1133 = X357-of-A'B'C'.
< 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) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1133 = BURGESS POINT (0.5521, 0.2022) A B C X1133 (a, b, c) = (5, 6, 7) | AB = 1

X1133 (BURGESS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: \(\sin(\pi /3 - A/3)/\sin(\pi /3 + A/3) : :\)
Search Value (6, 9, 13): \(2.415891690550\)
▸ Functional Relations1 relation

Notes

A. G. Burgess, “Concurrency of lines joining vertices of a triangle to opposite vertices of triangles on its sides,”Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 58-64; page 63.