From: Bill Dubuque <wgd@berne.ai.mit.edu>
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,
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
==============================================================================

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.

>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

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) 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

http://www.emis.de/cgi-bin/MATH-item?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, 130-133.  MR 90k:51032

http://www.emis.de/cgi-bin/MATH-item?704.51013

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

http://www.emis.de/cgi-bin/MATH-item?727.51006

>
>[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


http://www.emis.de/cgi-bin/MATH-item?844.51011



BTW two obituaries for Thebault:

Deaux, R.: Victor Thebault (1882-1960). 
Mathesis 69 (1961) 377-395.

Guillotin, M.R.: Victor Thebault (1882-1960). 
Scripta Math. 25 (1961) 331-333.


And a little problem:

In what triangle(s) the three circles of the Thebault's problem are congruent?



Antreas