Question 52
Consider a circle $\omega_1$ centered at $O_1$ and a circle $\omega_2$ centered at $O_2$ passes through $O_1$ and intersects $\omega_1$ at $A$ and $B$. Let $\ell$ be the tangent line from $A$ to $\omega_1$. A circle passes through $O_1$ and $O_2$ and its center lies on $\ell$, intersects $\omega_2$ for the second time at $P$. Prove that the reflection of $P$ with respect to $\ell$ lies on $\omega_1$.