# Morley's Miracle

Leo Giugiuc's Proof

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

In $\Delta QAC,$ we have $\displaystyle\frac{AQ}{\sin\frac{C}{3}}=\frac{AC}{\sin\frac{A+C}{3}};$ but $AC=2R\cdot\sin B,$ where $R$ is the circumradius of $\Delta ABC.$ It follows that

$\displaystyle\frac{AQ}{\sin\frac{C}{3}}=\frac{2R\cdot\sin B}{\sin\frac{A+C}{3}}=8R\cdot\sin\frac{B}{3}\sin\frac{A+C}{3}\sin\frac{\pi +B}{3},.$

implying $\displaystyle AQ=8R\cdot\sin\frac{B}{3}\sin\frac{C}{3}\sin\frac{\pi +C}{3}.$ But the numbers $\displaystyle\frac{A}{3},$ $\displaystyle\frac{\pi +B}{3},$ and $\displaystyle\frac{\pi +C}{3}$ are strictly positive and their sum is $\pi,$ so there is a triangle $XYZ,$ with $\displaystyle X=\frac{A}{3},$ $\displaystyle Y=\frac{\pi +B}{3},$ and $\displaystyle Z=\frac{\pi +C}{3}$ and the circumradius of $\displaystyle\frac{1}{2}.$

In $\Delta XYZ,$ $\displaystyle YZ=\sin\frac{A}{3},$ $\displaystyle XZ=\sin\frac{\pi +B}{3},$ and $\displaystyle XY=\sin\frac{\pi +C}{3};$ but $\displaystyle\frac{AQ}{XZ}=\frac{AR}{XY}=8R\cdot\sin\frac{B}{3}\cdot\sin\frac{C}{3}$ and $\displaystyle\angle QAR=X=\frac{A}{3},$ so that $\Delta AQR$ is similar to $\Delta XZY$ and, therefore, $\displaystyle\frac{QR}{YZ}=8R\cdot\sin\frac{B}{3}\cdot\sin\frac{C}{3}$ which finally gives $\displaystyle QR=8R\cdot\sin\frac{A}{3}\cdot\sin\frac{B}{3}\cdot\sin\frac{C}{3},$ a symmetric function of the angles of $\Delta ABC.$ This leads to $QR=PQ=PR.$

### 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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