Question 80
Let $ABC$ be an acute triangle. Suppose a circle $\Gamma_1$, with centre $O_1$, touches the sides $BC$ produced at $E$, $AC$ produced at $G$, and $AB$ at $C'$. Suppose also that another circle $\Gamma_2$, with centre $O_2$, touches the sides $AB$ produced at $H$, $BC$ produced at $F$, and $AC$ at $B'$. Let the extensions of $EG$ and $FH$ intersect at $P$. Prove that $PA \perp BC$.