University of Waterloo.
30 December 1975
Dear Sir,
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,
Edsger W.Dijkstra