Question 16
Let circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent of the two circles closest to the point $D$ touches $\Gamma_1$ at $A$ and $\Gamma_2$ at $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the middle of the line segment $BC$. Prove that $\angle DPM = \angle BDC$.