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Topic: A Problem
Replies: 5   Last Post: May 3, 1999 7:29 AM

 Messages: [ Previous | Next ]
 Bill Dubuque Posts: 156 Registered: 12/6/04
Re: A Problem
Posted: May 2, 1999 7:03 PM

xpolakis@hol.gr (Antreas P. Hatzipolakis) writes:
|
| In a triangle ABC, let D be a point on BC lying between B and C.
| If K_1,K_2 are the centers of the circles situated on the same side of
| BC as A and tangent to BC, to AD and internally to the circumcircle of
| ABC, prove that K_1 K_2 goes through the incenter of ABC.
| (V. Thebault, 1938)

This is the famous "Thebault Problem" [Amer. Math. Monthly 45 (1938), no. 7,
482-483, Advanced Problem 3887]. The first published solution was in 1983
by K. B. Taylor, who published [1] only an outline of his 24 page solution.
In 1986 G. Turnwald published [2] a complete 2 page trigonometric proof,
followed by a synthetic solution [3] by R. Stark. See [4] and [5] for
recent work and generalizations. All this info and more can be obtained
by searching MathSciNet at http://www.ams.org/msnmain?screen=Home

-Bill Dubuque

[1] K. B. Taylor. Three circles with collinear centres, Solution
of Advanced Problem 3887, Amer. Math. Monthly 90 (1983) 486-487.

[2] Turnwald, Gerhard. Ueber eine Vermutung von Thebault. (German) [On a
conjecture of Thebault] Elem. Math. 41 (1986), no. 1, 11-13. MR 88c:51018

[3] Stark, R. Eine weitere Losung der Thebault'schen Aufgabe. (German)
[Another solution of Thebault's problem]
Elem. Math. 44 (1989), no. 5, 130-133. MR 90k:51032

[4] Demir, H.; Tezer, C. Reflections on a problem of V. Thebault.
Geom. Dedicata 39 (1991), no. 1, 79-92. MR 92h:51029

[5] Rigby, John F. Tritangent centres, Pascal's theorem and Thebault's
problem. J. Geom. 54 (1995), no. 1-2, 134-147. MR 96h:51014