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.



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.

Bankoff's conundrum></p>

<p>Since angles PBQ, PQB, and DCP are equal, we have</p>

<p class= 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 YQ = ZB; and since PQ = PB, it follows that PZ = PY. Then, since X is the incenter of triangle BCP, PX bisects the angle BPC, and triangles XYP and PZX are congruent. So ZX = XY. Similarly, it can be shown that ZY = ZX = XY.

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

  1. Doodling and Miracles
  2. Morley's Pursuit of Incidence
  3. Lines, Circles and Beyond
  4. On Motivation and Understanding
  5. Of Looking and Seeing

Backward proofs

  1. J.Conway's proof
  2. D. J. Newman's proof
  3. B. Bollobás' proof
  4. G. Zsolt Kiss' proof
  5. Backward Proof by B. Stonebridge
  6. Morley's Equilaterals, Spiridon A. Kuruklis' proof
  7. J. Arioni's Proof of Morley's Theorem

Trigonometric proofs

  1. Bankoff's proof
  2. B. Bollobás' trigonometric proof
  3. Proof by R. J. Webster
  4. A Vector-based Proof of Morley's Trisector Theorem
  5. L. Giugiuc's Proof of Morley's Theorem
  6. Dijkstra's Proof of Morley's Theorem

Synthetic proofs

  1. Another proof
  2. Nikos Dergiades' proof
  3. M. T. Naraniengar's proof
  4. An Unexpected Variant
  5. Proof by B. Stonebridge and B. Millar
  6. Proof by B. Stonebridge
  7. Proof by Roger Smyth
  8. Proof by H. D. Grossman
  9. Proof by H. Shutrick
  10. Original Taylor and Marr's Proof of Morley's Theorem
  11. Taylor and Marr's Proof - R. A. Johnson's Version
  12. Morley's Theorem: Second Proof by Roger Smyth
  13. Proof by A. Robson

Algebraic proofs

  1. Morley's Redux and More, Alain Connes' proof

Invalid proofs

  1. Bankoff's conundrum
  2. Proof by Nolan L Aljaddou
  3. Morley's Theorem: A Proof That Needs Fixing

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