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, 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