### Morley's Miracle

H. D. Grossman'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 was published in *The American Mathematical Monthly*, Vol. 50, No. 9 (Nov., 1943), p. 552.

Let the triangle have base BC and angles 3α, 3β,3γ. Let BDK, BF, CDH, CE be angle trisectors. E is determined by making

∠EDF = 360° - (180° - β - γ) - (60° + β) - (60° + γ) = 60°.

Also

∠BFD = 180° - (60° + β + γ) = 60° + α.

Similarly, ∠CED = 60° + α.

Since D is equidistant from BF and CE,

∠1 = (60° + α) - (β - γ) = 60° - β.

Similarly,

∠2 = 60° - γ.

Through F draw line r making

∠3 = (60° + α) - (60° - β) = α + β

and

∠mr = (α + β) - β = α.

Similarly,

∠sn = (α + γ) - γ = α.

Further,

∠mn = (180° - 3β - 3γ) = 3α.

It remains only to prove that the lines m, n, r, and s converge to a point. The line KF joins the vertices of two isosceles triangles and therefore bisects ∠K. Then in triangle mBKs the bisector of ∠ms passes through F and being parallel to r, coincides with it. Similarly in triangle rHCn the bisector of ∠rn passes through E and being parallel to s, coincides with it.

### 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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