A Trigonometrical Approach to Morley‘s Observation

Ioannis Gasteratos; Spiridon Kuruklis; Thedore Kuruklis

doi:10.4067/S0719-06462017000200073

Authors

Ioannis Gasteratos Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA.
Spiridon Kuruklis Eurobank, Group Information and IT Security, 14234 Athens, Greece.
Thedore Kuruklis Theoklitos, 17672 Kallithea, Greece.

DOI:

https://doi.org/10.4067/S0719-06462017000200073

Keywords:

Angle trisection, proximal trisector, triangle trisectors, Morley‘s theorem, Morley triangle, Morley‘s magic, Morley‘s miracle, Morley‘s mystery

Abstract

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.