# Problem A. 368. (March 2005)

**A. 368.** The interior bisector of the angle *A* of the triangle *ABC* cuts the incircle *c* at two points; the one closer to *A* is denoted by *O*_{A}. Define the points *O*_{B} and *O*_{C} similarly on the internal bisectors of the angles *B* and *C*, respectively. The circle *c*_{A} is drawn about *O*_{A} and it is touching the sides *AB* and *CA*. Similarly, *c*_{B} is about *O*_{B} and it is touching the sides *BC* and *AB*, finally the circle *c*_{C} is about *O*_{C} and it is touching the sides *CA* and *BC*. Taken any two of the circles *c*_{A}, *c*_{B} and *c*_{C} consider their common external tangents different from the corresponding sides of the triangle, respectively. Prove that these three tangents are concurrent.

(5 pont)

**Deadline expired on April 15, 2005.**

### Statistics:

7 students sent a solution. 5 points: Jankó Zsuzsanna, Kiss-Tóth Christian, Mánfay Máté, Molnár 999 András, Paulin Roland, Strenner Balázs. 4 points: Pálinkás Csaba.

Problems in Mathematics of KöMaL, March 2005