Problem A. 564. (May 2012)
A. 564. Let k be the incircle in the triangle ABC, which is tangent to the sides AB, BC, CA at the points C_{0}, A_{0} and B_{0}, respectively. The angle bisector starting at A meets k at A_{1} and A_{2}, the angle bisector starting at B meets k at B_{1} and B_{2}; AA_{1}<AA_{2} and BB_{1}<BB_{2}. The circle k_{1}k is tangent externally to the side CA at B_{0} and it is tangent to the line AB. The circle k_{2}k is tangent externally to the side BC at A_{0}, and it is tangent to the line AB. The circle k_{3} is tangent to k at A_{1}, and it is tangent to k_{1} at point P. The circle k_{4} is tangent to k at B_{1}, and it is tangent to k_{2} at point Q. Prove that the radical axis between the circles A_{1}A_{2}P and B_{1}B_{2}Q is the angle bisector starting at C.
(5 pont)
Deadline expired on 11 June 2012.
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