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.
Since angles PBQ, PQB, and DCP are equal, we have
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.
- J.Conway's proof
- D. J. Newman's proof
- Bankoff's proof
- B. Bollobás' proof
- Another proof
- Nikos Dergiades' proof
- G. Zsolt Kiss' proof
- M. T. Naraniengar's proof
- Doodling and Miracles
- Morley's Pursuit of Incidence
- Lines, Circles and Beyond
- On Motivation and Understanding
- Bankoff's conundrum
- Of Looking and Seeing
- Morley's Redux and More, Alain Connes' proof
- An Unexpected Variant
- Proof by B. Stonebridge and B. Millar
- Proof by B. Stonebridge
- Proof by Nolan L Aljaddou
- Proof by Roger Smyth
Copyright © 1996-2009 Alexander Bogomolny
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