CTK Insights

In 1938, the French V. Thébault proposed a problem that, in the English speaking world waited for a solution for 45 years until one was supplied by K. B. Taylor in 1983. Taylor's solution stretched over 24 printed pages. In Dutch, a solution has been published by H. Streefkerk's 10 years earlier. In 2003, Jean-Louis Ayme reported a proof based on a lemma from 1905 by the Japanese Y. Sawayama. Curiously, the papers by Thébault , Taylor and Sawayama all have been published in The American Mathematical Monthly, a publication of the Mathematical Association of America. In fact, Sawayama's paper is dedicated to a solution to what became known as Thébault 's problem some 33 years later. Here is the problem:

Let AM be a cevian in ΔABC, M between B and C. Construct two circles that touch BC, AM and the circumcircle of ΔABC and denote their centers as P and Q. Let I be the incenter of ABC. Then P, I, Q are collinear.

As often the case in mathematics, a curious property of a configuration seldom comes along. One such curiousity concerning Thébault 's problem I came across fairly recently. Among possible positions of point M on BC (at least) one is special. If M is the point of tangency of the excircle opposite vertex A, then the three circles in Thébault's configuration are congruent:

Indeed the result I found was a little different. The two circles tangent to AM and BC were tangent to the circumcircle externally and there centers were collinear with an excenter of ΔABC. The three became congruent when M was the point of tangency of the incircle! However, the two facts are equivalent due to the extraversion principle introduced by J. Conway. Roughly speaking, extraversion is similar to the principle of duality in projective geometry where a statement remains true after swapping all occurencies of the words point and line. According to the principle of extraversion a statement remains true after swapping the words incircle and excircle.

References

1. Thébault's Problem III
2. Y. Sawayama's Lemma
3. Y. Sawayama's Theorem
4. Thébault's Problem III, Proof (J.-L. Ayme)
5. Circles Tangent to Circumcircle

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