Question 145
Given circles $\omega_1, \omega_2$ with centers $O_1, O_2$, externally tangent to each other at the point $P$. A line $l$ tangent to $\omega_1$ meets $\omega_2$ at points $A, B$. A line through $B$ perpendicular to $l$ and the line $O_1A$ meet at $C$. The line $PC$ and the segment $AB$ meet at $Q$, and the line $O_1Q$ and the segment $BC$ meet at $D$. Prove that $D$ is the midpoint of the segment $BC$.
(proposed by B. Battsengel)
(proposed by B. Battsengel)