On the Morley's triangle

TL;DR

This paper thoroughly investigates Morley's trisector theorem, proving that the intersections of angle trisectors form an equilateral triangle only when the lines are true trisectors, using elementary trigonometry and polynomial systems.

Contribution

It provides a complete proof of Morley's theorem and introduces a potentially new trigonometric identity through systematic mathematical analysis.

Findings

Intersections form an equilateral triangle only with true trisectors.

Elementary trigonometry and polynomial equations are sufficient for the proof.

A new trigonometric identity is derived as a byproduct.

Abstract

We give a complete investigation of Morley's trisector theorem. If the intersections of the half lines starting from the adjacent vertices of a triangle form an equilateral triangle for an arbitrary triangle, then the half lines are the angle trisectors. To derive the result we use elementary trigonometry, Taylor series expansions, and solve systems of polynomial equations step by step. As a byproduct we get a probably new trigonometric identity.