Question 161
Let $ABC$ be a triangle and let $P$ be a point in the interior of this triangle. Let $\omega_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally. Let $\Gamma_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ externally and tangent to the circumcircle of $ABC$ at $A_2$ internally. Define $B_1, B_2, C_1, C_2$ similarly. Let $O$ be the circumcentre of $ABC$. Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ and $OP$ are concurrent.