Question 4
Let $A,B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of $AB$. Let $O_1$ be the circle tangent to line $AB$ at $P$ and tangent to circle $O$. Let $\ell$ be the tangent line, different from $AB$, to $O_1$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $\ell$ and $O$. Let $Q$ be the midpoint of $BC$ and $O_2$ be the circle tangent to line $BC$ at $Q$ and tangent to segment $AC$. Prove that circle $O_2$ is tangent to circle $O$.
