# Morley's Miracle

Bankoff's Conundrum

Professor McWorter sent me a one page paper he came across while rummaging through some old stuff. The paper appeared in __Eureka__, Vol. 2, no. 8, October, 1976, p. 162. It's reproduced below as accurately as possible. Two footnotes are either Bankoff's or the journal editor's. The second one is especially noteworthy.

A DIRECT GEOMETRICAL PROOF OF MORLEY'S THEOREM

EUCLIDE PARACELSO BOMBASTO UMBIGIO, Guyazuela

*MORLEY"S THEOREM. The intersections of the adjacent internal angle trisectors of a triangle are the vertices of an equilateral triangle.*

*Proof* ^{1}. Extend BZ and CY to meet at P (see figure). On the segment PC, let PQ = PB, and let L be the projection of Z on BQ. Construct CD parallel to BQ and let M, N denote projections of Y, Q on CD.

YM/YC = ZL/ZB = QN/QC or (YM - ZL)/(YC - ZB) = QN/QC.

But YM - ZL = QN; hence YC - ZB = QC. But YC - YQ = QC. Therefore

*Q.E.D. et N.F.C.* ^{2}

(^{1})
This proof was communicated by the renowned problemist, Professor Euclide Paracelso Bombasto Umbigio, Guyazuela, to Dr. LEON BANKOFF, Los Angeles, California, who kindly translated it for us. The original proof was written in Esperanto, which Dr. Bankoff speaks like a native. Professor Umbigio is known primarily as a numerologist; this is one of his rare excursions in geometry.

(^{2}) *N.F.C.* is the abbreviation of *Ne Fronti Crede*, the Latin equivalent of "Don't believe everything you see." Dr. Bankoff says that, to avoid embarrassment for the good professor, he took the liberty of adding *N.F.C* to his *Q.E.D.* Those familiar with Professor Umbigio's published papers will recognize the need for this minor addendum.

### 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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