ELMO SL 2011 G2

Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$ , $\omega_1,\omega$ are internally tangent at $A$ , and $\omega,\omega_2$ are internally tangent at $B$ . Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$ , respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$ , prove that $\angle{O_1XP}=\angle{O_2XP}$ .

David Yang.