Question 88
Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$. Lines $AP$, $BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A$, $B$, $C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$. The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2$, $BB_2$ and $CC_2$ are concurrent.