## Morley's Miracle

Nikos Dergiades' proof

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) ΔABC is equal to 90° + A/2.

### Lemma 2

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

### Proof of Morley's Theorem

Given ΔABC, with angles A, B, C. If

On the sides of an arbitrary equilateral triangle A_{1}B_{1}C_{1} we construct outwardly the triangles A'B_{1}C_{1}, A_{1}B'C_{1}, and A_{1}B_{1}C' such that the lines B_{1}C' and B'C_{1} are symmetric with respect to the perpendicular bisector of B_{1}C_{1} and also _{1}A_{1} = ∠C'B_{1}A_{1} = x

It is obvious that the angles of the three triangles A'B_{1}C_{1}, A_{1}B'C_{1}, A_{1}B_{1}C' at A', B', C' are A/3, B/3, C/3, respectively.

If A_{2} is the intersection of B'C_{1} and C'B_{1}, then the triangle A_{2}B_{1}C_{1} is isosceles and A_{2}A_{1} is the bisector of _{2} = ∠B'A_{2}C'.

By Lemma 1, we have that x + 60 = ∠B'C_{1}B_{1} = 90° + A_{2}/2.

Since _{1}C' = x + 60° = 90° + A_{2}/2_{1} is the incenter of triangle A_{2}B'C', and similarly B_{1}, C_{1} are the incenters of triangles B_{2}C'A', C_{2}A'B' respectively. Hence _{1} = ∠C_{1}A'B_{1} = ∠B_{1}A'C' = A/3,_{1}B_{1}C_{1} that is equilateral.

### Morley's Miracle

#### On Morley and his theorem

- Doodling and Miracles
- Morley's Pursuit of Incidence
- Lines, Circles and Beyond
- On Motivation and Understanding
- Of Looking and Seeing

#### Backward proofs

- J.Conway's proof
- D. J. Newman's proof
- B. Bollobás' proof
- G. Zsolt Kiss' proof
- Backward Proof by B. Stonebridge
- Morley's Equilaterals, Spiridon A. Kuruklis' proof
- J. Arioni's Proof of Morley's Theorem

#### Trigonometric proofs

- Bankoff's proof
- B. Bollobás' trigonometric proof
- Proof by R. J. Webster
- A Vector-based Proof of Morley's Trisector Theorem
- L. Giugiuc's Proof of Morley's Theorem
- Dijkstra's Proof of Morley's Theorem

#### Synthetic proofs

- Another proof
- Nikos Dergiades' proof
- M. T. Naraniengar's proof
- An Unexpected Variant
- Proof by B. Stonebridge and B. Millar
- Proof by B. Stonebridge
- Proof by Roger Smyth
- Proof by H. D. Grossman
- Proof by H. Shutrick
- Original Taylor and Marr's Proof of Morley's Theorem
- Taylor and Marr's Proof - R. A. Johnson's Version
- Morley's Theorem: Second Proof by Roger Smyth
- Proof by A. Robson

#### Algebraic proofs

#### Invalid proofs

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