  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.
