Question 4
Circle $\Gamma$ is tangent to sides $AB$ and $AC$ of triangle $ABC$ at $E$ and $F$, respectively. Let $X$ be the intersection of $BF$ and $EC$, and let $H$ be the intersection of $\Gamma$ with $AX$. Let $Z$ and $T$ be the intersections of $EH$ and $FH$ with $BC$, respectively. The lines $ET$ and $FZ$ intersect at $Q$. Prove that the point $Q$ is on the line $AX$.