Question 55
A circle $\omega$ with center $P$ intersects the sides $BC, CA, AB$ of a triangle $ABC$ at points $A_1$ and $A_2, B_1$ and $B_2, C_1$ and $C_2$, respectively. Let $A'$ be the circumcenter of the triangle $A_1A_2P$, $B'$ be the circumcenter of the triangle $B_1B_2P$ and $C'$ be the circumcenter of the triangle $C_1C_2P$. Prove that the lines $AA', BB'$ and $CC'$ are concurrent.