From: Bill Dubuque
Subject: Re: A Problem
Date: 02 May 1999 19:03:37 0400
Newsgroups: sci.math
Keywords: Thebault Problem (circles and triangles)
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,
482483, 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) 486487.
[2] Turnwald, Gerhard. Ueber eine Vermutung von Thebault. (German) [On a
conjecture of Thebault] Elem. Math. 41 (1986), no. 1, 1113. 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, 130133. MR 90k:51032
[4] Demir, H.; Tezer, C. Reflections on a problem of V. Thebault.
Geom. Dedicata 39 (1991), no. 1, 7992. MR 92h:51029
[5] Rigby, John F. Tritangent centres, Pascal's theorem and Thebault's
problem. J. Geom. 54 (1995), no. 12, 134147. MR 96h:51014
==============================================================================
From: xpolakis@hol.gr (Antreas P. Hatzipolakis)
Subject: Re: A Problem
Date: 3 May 1999 02:15:36 GMT
Newsgroups: sci.math
Bill Dubuque wrote:
>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,
Now that you Bill revealed the Problem's difficulty nobody will try... :)
Who knows... had someone tried probably would be able to give a simple
solution.....
Anyway, Thanks for your response.
>482483, 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
Reviews in ZfM for those with no access to MathSciNet:
>
>[1] K. B. Taylor. Three circles with collinear centres, Solution
>of Advanced Problem 3887, Amer. Math. Monthly 90 (1983) 486487.
>
>[2] Turnwald, Gerhard. Ueber eine Vermutung von Thebault. (German) [On a
>conjecture of Thebault] Elem. Math. 41 (1986), no. 1, 1113. MR 88c:51018
http://www.emis.de/cgibin/MATHitem?583.51016
>
>[3] Stark, R. Eine weitere Losung der Thebault'schen Aufgabe. (German)
>[Another solution of Thebault's problem]
>Elem. Math. 44 (1989), no. 5, 130133. MR 90k:51032
http://www.emis.de/cgibin/MATHitem?704.51013
>
>[4] Demir, H.; Tezer, C. Reflections on a problem of V. Thebault.
>Geom. Dedicata 39 (1991), no. 1, 7992. MR 92h:51029
http://www.emis.de/cgibin/MATHitem?727.51006
>
>[5] Rigby, John F. Tritangent centres, Pascal's theorem and Thebault's
>problem. J. Geom. 54 (1995), no. 12, 134147. MR 96h:51014
http://www.emis.de/cgibin/MATHitem?844.51011
BTW two obituaries for Thebault:
Deaux, R.: Victor Thebault (18821960).
Mathesis 69 (1961) 377395.
Guillotin, M.R.: Victor Thebault (18821960).
Scripta Math. 25 (1961) 331333.
And a little problem:
In what triangle(s) the three circles of the Thebault's problem are congruent?
Antreas