# The Fundamental Proof of Morley's Trisector Theorem

### Nolan L Aljaddou

April 26, 2009

All triangle endpoints are formed by the intersection of three distinct lines. That is, a triangle with three side lines A, B, and C is generated as follows:

The trisection of a triangle's angles divides them each into three equal values, and one third of each separate original angle respectively may be denoted as a, b and c. Since their original total sum is 180°

Since each angle of an equilateral triangle is 60°, each of its angles can be divided into three parts which are equal to a, b, and c - which then form a union to form the angle. Therefore, the equilateral triangle's endpoints themselves are generated by the intersection of the union of these three angles, three times. That is:

QED

**Note by AlexB**: the above is a version of the original message which can be found - along with a record of the intervening correspondence - on a separate page.

### 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

- Bankoff's conundrum
- Proof by Nolan L Aljaddou
- Morley's Theorem: A Proof That Needs Fixing

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