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Thébault's Problem III: What Is It?
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Copyright © 1996-2008 Alexander Bogomolny
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Thébault's Problem III
The applet suggests the following theorem (Victor Thébault, 1938):
Let AM be a cevian in  ABC, M between B and C. Construct two circles that touch BC, AM and the circumcircle of ABC and denote their centers as P and Q. Let I be the incenter of ABC. Then P, I, Q are collinear.
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This is a problem posed by the French mathematician Victor Thébault (1882-1960) in 1938 (Amer. Math. Monthly 45 (1938), no. 7, 482-483, Advanced Problem 3887). In the English speaking world it was believed that the first solution was found by K. B. Taylor in 1983, 45 years later. (Taylor's solution occupied 24 printed pages. In [Taylor] only a summary was presented.) This much has been asserted as late as 2001 by R. Shail, himself an author of an analytic proof (Amer. Math. Monthly 108 (2001), no. 4, 319-325). However, unbeknownst to the English, in the parallel universe of the Dutch language events evolved at a faster rate. To the Dutch it took only 35 years to solve the problem. The following is an excerpt from a discussion at the geometry-college newgroup
Antreas P. Hatzipolakis > The problem, published in 1938, remained Some articles started with Dr. H. Streefkerk's solution of the Thebault problem in 1973. It can be found in issues of july 1973, november 1973, january 1974 and march 1974 of the Dutch magazine "Nieuw Tijdschrift voor Wiskunde". The articles are all in Dutch. A very nice is the short one of Prof. dr. G. R. Veldkamp, it's a real beauty. It is a complete solution of Thebault's problem with some more in the issue of november 1973 p. 86-89. Veldkamp starts from an extension of Ptolemy's theorem as follows: Instead of 4 points on a circle OMEGA, take 3 points together with a circle GAMMA tangent to OMEGA. Replace 3 of the 6 distances involved in Ptolemy's theorem by the length of the tangent from 3 points to GAMMA. Best Regards, Frans Gremmen, University of Nijmegen, The Netherlands. |
The references below have been garnered by Bill Dubuque
Reference
Four years later a more complete history along with a synthetic solution has been published by the Frenchman Jean-Louis Ayme:
who discovered a theorem by Y. Sawayama in an early issue of the American Mathematical Monthly (v 12 (1905), no. 12, 222-224). Sawayama's statement is in fact more general than Thébault's and the 2 page proof is entirely synthetic and elementary.
- Thébault's Problem I
- Thébault's Problem II
- Thébault's Problem III
- Y. Sawayama's Lemma
- Y. Sawayama's Theorem
- Thébault's Problem III, Proof (J.-L. Ayme)
Copyright © 1996-2008 Alexander Bogomolny
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