The proof has been published (in Greek) in the bulletin of the department (Central Makedonia of Greece) of the Greek Mathematical Society "Diastasi": Nikolaos Dergiades, "A simple geometric proof of Morley's theorem", Diastasi 1991 is. 1-2 p. 37-38 Thessaloniki-Greece.

Lemma 1

The external angle B of an isosceles (AB = AC) ABC is equal to 90^o + A/2.

Lemma 2

The incenter I of ABC lies on the bisector of an angle, e.g A, and sees the opposite side BC with angle BIC = 90^o + A/2 and conversely.

Proof of Morley's Theorem

Given ABC, with angles A, B, C. If x = 60^o + A/3, y = 60^o + B/3, z = 60^o + C/3, then x + y + z + 120^o = 360^o.

On the sides of an arbitrary equilateral triangle A₁B₁C₁ we construct outwardly the triangles A'B₁C₁, A₁B'C₁, and A₁B₁C' such that the lines B₁C' and B'C₁ are symmetric with respect to the perpendicular bisector of B₁C₁ and also B'C₁A₁ = C'B₁A₁ = x and similarly the others as seen in the figure.

It is obvious that the angles of the three triangles A'B₁C₁, A₁B'C₁, A₁B₁C' at A', B', C' are A/3, B/3, C/3, respectively.

If A₂ is the intersection of B'C₁ and C'B₁, then the triangle A₂B₁C₁ is isosceles and A₂A₁ is the bisector of A₂ = B'A₂C'.

By Lemma 1, we have that x + 60 = B'C₁B₁ = 90^o + A₂/2.

Since B'A₁C' = x + 60^o = 90^o + A₂/2, by Lemma 2 we conclude that A₁ is the incenter of triangle A₂B'C', and similarly B₁, C₁ are the incenters of triangles B₂C'A', C₂A'B' respectively. Hence B'A'C₁ = C₁A'B₁ = B₁A'C' = A/3, or, finaly, A' = A, B' = B, C' = C which means that the triangles ABC, A'B'C' are similar and the trisectors of the angles of ABC form a triangle similar to A₁B₁C₁ that is equilateral.

Morley's Miracle

1. J.Conway's proof
   • Remarks on J.Conway's proof
2. D.J.Newman's proof
3. Bankoff's proof
4. Another proof
5. Nikos Dergiades' proof
6. G. Zsolt Kiss' proof
7. M. T. Naraniengar's proof
8. Doodling and Miracles
9. Morley's Pursuit of Incidence
10. Lines, Circles and Beyond
11. On Motivation and Understanding
12. Bankoff's Conundrum
13. Morley's Redux and More, Alain Connes' proof
14. An Unexpected Variant

Copyright © 1996-2008 Alexander Bogomolny