ELMO SL 2011 G3

Let $ABC$ be a triangle. Draw circles $\omega_A$ , $\omega_B$ , and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$ , and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$ . Define $P_B$ and $P_C$ similarly. Prove that $AP_A$ , $BP_B$ , and $CP_C$ are concurrent.

Tom Lu.