An open letter to Ross Honsberger.
University of Waterloo.
30 December 1975
the other day I encountered your delightful booklet “Mathematical Gems”. On account of Chapter 8, I concluded that you might be interested in the following proof of Morley’s Theorem “The adjacent pairs of the trisectors of a triangle always meet at the vertices of an equilateral triangle.”
Choose α, β & γ > 0 such that α + β + γ = 60°. Draw an equilateral triangle XYZ and construct the triangles AXY and BXZ with the angles as indicated. Because ∠AXB = 180° – (α+β), it follows that, if ∠BAX = α+x, ∠ABX = β – x. Using the rule of sines three times (in △AXB, △AXY, and △BXZ), we deduce
|sin(α+x)||=||BX||=||XZ . sin(60+γ)/sin(β)||=||sin(α)|
|sin(β–x)||AX||XY . sin(60+γ)/sin(α)||sin(β)|
Because in the range considered, this equation has a left-hand-side which is a monotonically increasing function of x (on account of the monotonicity of sin(φ) in the first quadrant) we conclude x = 0. Thus Morley’s Theorem is proved without any additional lines. I found this proof in the early sixties, but am afraid that I did not publish it. Yours ever,