HMMT 2003 G9

In triangle $A B C, \angle A B C=50^{\circ}$ and $\angle A C B=70^{\circ}$. Let $D$ be the midpoint of side $B C$. A circle is tangent to $B C$ at $B$ and is also tangent to segment $A D$; this circle instersects $A B$ again at $P$. Another circle is tangent to $B C$ at $C$ and is also tangent to segment $A D$; this circle intersects $A C$ again at $Q$. Find $\angle A P Q$ (in degrees).