## Morley's Miracle

R. J. Webster's Proof

### Theorem

The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. |

### Proof

This proof appeared in *Mathematics Magazine*, Vol. 43, No. 4 (Sep., 1970), pp. 209-210.

Let ABC be a triangle with inradius r and circumradius R, and let the adjacent trisectors of angles A, B, C meet in A', B', C' as illustrated below. Morley's theorem states that ΔA'B'C' is equilateral. The proofs of this theorem, which are usually given, do not include calculation of the side of this triangle.

In this note we prove Morley's theorem by showing that the side of this triangle is

In any triangle ABC, we have a = 2R sin(A), etc. Let

A'B | = 2R sin(3α) sinγ / sin(β + γ) | |

= 2R (4sinα sin(60° + α) sin(60° - α) sinγ) / sin(60° - α) | ||

= 8R sinα sinγ sin(60° + α). | ||

Similarly, BC' = 8R sinα sinγ sin(60° + γ). Consider now triangle DEF shown above, where

**Editorial Note**: Professor C. N. Mills of Illinois State University at Normal as a *tour de force* found above expression for a side of the Morley triangle by a straightforward use of elementary Cartesian analysis! His complete proof required some twenty 8½×11 sheets of paper.

### 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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