CMO 2012 Problem 1

In the triangle $ABC$ , $\angle A$ is biggest. On the circumcircle of $\triangle ABC$ , let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$ . The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$ , the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$ . $c_1$ and $c_2$ intersect at $A$ and $P$ . Prove that $AP$ bisects $\angle BAC$ .