firstname.lastname@example.org (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  only an outline of his 24 page solution. In 1986 G. Turnwald published  a complete 2 page trigonometric proof, followed by a synthetic solution  by R. Stark. See  and  for recent work and generalizations. All this info and more can be obtained by searching MathSciNet at http://www.ams.org/msnmain?screen=Home
 K. B. Taylor. Three circles with collinear centres, Solution of Advanced Problem 3887, Amer. Math. Monthly 90 (1983) 486-487.
 Turnwald, Gerhard. Ueber eine Vermutung von Thebault. (German) [On a conjecture of Thebault] Elem. Math. 41 (1986), no. 1, 11-13. MR 88c:51018
 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
 Demir, H.; Tezer, C. Reflections on a problem of V. Thebault. Geom. Dedicata 39 (1991), no. 1, 79-92. MR 92h:51029
 Rigby, John F. Tritangent centres, Pascal's theorem and Thebault's problem. J. Geom. 54 (1995), no. 1-2, 134-147. MR 96h:51014