ISL 2006 G3

Let $A B C D E$ be a convex pentagon such that $$\angle B A C=\angle C A D=\angle D A E \text { and } \angle A B C=\angle A C D=\angle A D E \text {. }$$ The diagonals $B D$ and $C E$ meet at $P$. Prove that the line $A P$ bisects the side $C D$.