prof.dr.Edsger W.Dijkstra, Burroughs Research Fellow
Plataanstraat 5
NUENEN - 4565
The Netherlands
EWD 544

An open letter to Ross Honsberger.

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.”

EWD544 figure 1

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(βx) 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