Problem A. 377. (September 2005)
A. 377. The inscribed circle of the triangle ABC touches the side AB at C_{1}, the side BC at A_{1}, and the side CA at B_{1}. It is known that the line segments AA_{1}, BB_{1} and CC_{1} pass through a common point. Let N denote that point. Draw the three circles that pass through N and touches two of the sides. Prove that the six points of tangency are concyclic.
(5 pont)
Deadline expired on October 17, 2005.
Solution. Use the notations of the Figure.
Circle B_{3}C_{2}N can be obtained by scaling the inscribed circle from vertex A. Triangle NB_{3}C_{2} is the image of triangle A_{1}B_{1}C_{1}, so the sides of these triangles are paralel, respectively. By similar scalings from points B and C we obtain that lines B_{2}N and NC_{3} are paralel to B_{1}C_{1}. This implies that line B_{2}C_{3} passes through point N and also this line is paralel to B_{1}C_{1}.
The perpendicular bisector of B_{1}C_{1} is the angle bisector AK. Since line segments B_{2}C_{3} and B_{3}C_{2} are paralel to B_{1}C_{1}, AK is also their perpendicular bisector.
Similarly, the line segments A_{2}B_{3} and A_{3}B_{2} share the common perpendicular bisector CK and segments A_{2}C_{3} and A_{3}C_{2} share BK.
All these three perpendicular bisectors pass through the incenter K, so KB_{2}=KC_{3}=KA_{2}=KB_{3}=KC_{2}=KA_{3} and points A_{2},A_{3},B_{2},B_{3},C_{2},C_{3} lie on a certain circle of center K.
Statistics:
23 students sent a solution.  
5 points:  Erdélyi Márton, Estélyi István, Fischer Richárd, Gyenizse Gergő, Hujter Bálint, Jankó Zsuzsanna, KisfaludiBak Sándor, Kónya 495 Gábor, Korándi Dániel, Kutas Péter, Lovász László Miklós, Molnár 999 András, Nagy 224 Csaba, Paulin Roland, Tomon István, Udvari Balázs, Ureczky Bálint, Viktor Simjanoski. 
0 point:  5 students. 
