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

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

(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

1. J.Conway's proof
   • Remarks on J.Conway's proof
2. D.J.Newman's proof
3. Bankoff's proof
4. Another proof
5. Nikos Dergiades' proof
6. G. Zsolt Kiss' proof
7. M. T. Naraniengar's proof
8. Doodling and Miracles
9. Morley's Pursuit of Incidence
10. Lines, Circles and Beyond
11. On Motivation and Understanding
12. Bankoff's Conundrum
13. Morley's Redux and More, Alain Connes' proof
14. An Unexpected Variant

Copyright © 1996-2008 Alexander Bogomolny