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Thébault's Theorem

Let DeltaABC be a triangle and D a point on the side BC. Let I be the incenter, P the center of the circle tangent to the circumcircle and segments AD and BD, Q the center of the circle tangent to the circumcircle and segments AD and CD. Then Thébault's theorem states that P, Q and I are collinear.

The problem was proposed by Thébault (1938) and apparently remained unsolved until the 1970s. It was therefore a surprise when Ayme (2003) discovered that in fact the problem was known and solved by Sawayama (1905).

This entry contributed by Floor van Lamoen

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References

Ayme, J.-L. "Sawayama and Thébault's Theorem." Forum Geom. 3, 225-229, 2003. http://forumgeom.fau.edu/FG2003volume3/FG200325.pdf.Demir, H. and Tezer, C. "Reflections on a Problem of V. Thébault." Geom. Dedicata 39, 79-92, 1991.Dijkstra-Kluyver, B. C. "Twee oude vraagstukken in een klap opgelost." Nieuw Tijdschr. Wisk. 61, 134-135, 1973-1974.Dijkstra-Kluyver, B. C. and Streefkerk, H. "Nogmaals het vraagstuk van Thébault." Nieuw Tijdschr. Wisk. 61, 172-173, 1973-1974.Dutta, S. "Thébault's Problem via Euclidean Geometry." Samayā 7, 2-7, 2001.English, B. J. "Solution of Problem 3887." Amer. Math. Monthly 110, 156-158, 2003.Gueron, S. "Two Applications of the Generalized Ptolemy Theorem." Amer. Math. Monthly 109, 362-370, 2002.Reyes, W. "An Application of Thébault's Theorem." Forum Geom. 2, 183-185, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200223index.html.Rigby, J. F. "Tritangent Centres, Pascal's Theorem and Thébault's Problem." J. Geom. 54, 134-147, 1995.Sawayama, Y. "A New Geometrical Proposition." Amer. Math. Monthly 12, 222-224, 1905.Shail, R. "A Proof of Thébault's Theorem." Amer. Math. Monthly 108, 319-325, 2001.Stark, R. "Eine weitere Lösung der Thébault'schen Aufgabe." Elem. Math. 44, 130-133, 1989.Streefkerk, H. "Waarom eenvoudig als het ook ingewikkeld kan?" Nieuw Tijdschr. Wisk. 60, 240-253, 1972-1973.Taylor, K. B. "Solution of Problem 3887." Amer. Math. Monthly 90, 482-487, 1983.Thébault, V. "Problem 3887. Three Circles with Collinear Centers." Amer. Math. Monthly 445, 482-283, 1938.Turnwald, G. "Über eine Vermutung von Thébault." Elem. Math. 41, 11-13 ,1986.Veldkamp, G. R. "Een vraagstuk van Thébault uit 1938." Nieuw Tijdschr. Wisk. 61, 86-89, 1973-1974.Veldkamp, G. R. "Solution to Problem 1260." Crux Math. 15, 51-53, 1989.

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Thébault's Theorem

Cite this as:

van Lamoen, Floor. "Thébault's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ThebaultsTheorem.html

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