# Morley's Miracle,

Bella Bollobás' Trigonometric Proof

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

For simplicity, let us write $x^{+}$ for $x+\pi/3.$ We claim that the angles of the six triangles are as in the diagram below:

Note that the assignment of angles is consistent with the requirement that the sum of angles in a triangle is $\pi.$ Simple calculations show that this is indeed so and that the angles in $\Delta PQR$ are each $60^{\circ}.$ We do know that the magnitudes of the angles $AQC,$ $BPC,$ and $ARB$ are as shown. To establish Morley's theorem we need to verify that other angles (those with a single $^{+})$ are distributed as in the diagram.

We assume that the circumradius of $\Delta ABC$ equals $1/2.$ This makes $BC=\sin 3\alpha,$ $AC=\sin 3\beta,$ and $AB=\sin 3\gamma.$ With the Law of Sines, we have, in $\Delta ABR,$ $\displaystyle AR=\frac{\sin\beta}{\sin\gamma^{++}}\sin 3\gamma$ and, in $\Delta ACQ,$ $\displaystyle AQ=\frac{\sin\gamma}{\sin\beta^{++}}\sin 3\beta.$

Hence, in order that in $\Delta ARQ,$ the angles be as claimed, implying, in particular, $\displaystyle\frac{AR}{AQ}=\frac{\sin\gamma^{+}}{\sin\beta^{+}},$ we need

$\sin 3\beta\sin\gamma\sin\gamma^{+}\sin\gamma^{++}=\sin 3\gamma\sin\beta\sin\beta^{+}\sin\beta^{++}.$

But this is indeed so because

$\sin 3x=4\sin x \sin x^{+} \sin x^{++}=3\sin x-4\sin^{3}x.$

### References

- B. Bollobás, The Art of Mathematics, Cambridge University Press, 2006, p. 126-127

### 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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