Question 161
Let triangle $ABC$ be right-angled at $A$. Let $M$ denote the midpoint of the segment $BC$. The point $D$ lies on the side $AC$ and satisfies $\overline{AD}=\overline{AM}$. The intersection point, different from $C$, of the circumcircles of the triangles $AMC$ and $BDC$ is called $P$. Prove that $CP$ bisects the angle of the triangle $ABC$ at $C$.