Simple trigonometrical arguments verify that in a triangle the trisectors, proximal to sides respectively, meet at the vertices of an equilateral triangle by showing that the length of each side is 8R times the sines of the angles between the sides of the triangle and the trisectors that determine it, where R is the radius of the circumcircle of the triangle. The 27 meeting points of the trisectors, proximal to a side, determine 18 such equilaterals, which in pairs share a vertex having two collinear sides and the third parallel. Hence these points are located 6 by 6 on three triples of parallel lines.
@article{1579,
title = {A Trigonometrical Approach to Morley's Observation},
journal = {CUBO, A Mathematical Journal},
volume = {19},
year = {2017},
language = {en},
url = {http://dml.mathdoc.fr/item/1579}
}
Gasteratos, Ioannis; Kuruklis, Spiridon; Kuruklis, Thedore. A Trigonometrical Approach to Morley’s Observation. CUBO, A Mathematical Journal, Tome 19 (2017) . http://gdmltest.u-ga.fr/item/1579/