# Morley's Theorem

### Roger Smyth

Morley's theorem asserts that in the diagram below the triangle XYZ is equilateral, whatever ΔABC.

In the next diagram, the unit of measure is the perpendicular DX, and the lengths of BX and CX are s and s'. E and F are points on BC where

What can be said of the diagram?

- α + β + γ = 60° (so ∠EXF = α)
- ∠CYA = ∠CLP' = 120° + β
- ∠TSB = ∠SPX = 60°
- AY = (AC/s')LP' (for, ΔCP'L is similar to ΔCAY)
- LP' = PS/ST (for ∠P'LY = ∠SPT = 60° - β)
- XR = RV = 2/s (for triangles BDX, PRX, PVR are similar)
- SU = s/2 (for ∠BSU = 60°)

Now, by (vi) and (vii),

2 ST = 2 SU + 2 UT = s + 2(s - 2/s) = 3s - 4/s.

Also, since triangles BQV and BDX are similar,

PQ = PV + VQ = 3 - 4/s²,

leading to 2 ST = s PQ.

Next, by (iv) and (v),

AY = (AC / s')(PS / ST) = 2 AC·PS / (ss' PQ).

But XE = 2s/PS as ΔPBS is isosceles and ∠SBP = 2(60° + β) = 2∠DEX. It follows that

XE·AY = 4 AC / (s' PQ)

and, by symmetry,

(XE·AY) / (XF·AZ) = [4 AC / (s' PQ)] [(s P'Q') / (4 AB)] = (s AC·P'Q') / (s' AB·PQ) = 1

because PQ(AB/s) = P'Q'(AC/s') is the altitude of ΔABC. However, this means

### References

- M. R. F. Smyth,
__MacCool's proof of Morley's Miracle__,*Irish Math. Soc. Bulletin*63 (2009), 63-66

### 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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