Question 6
Two circles $\Gamma_1$ and $\Gamma_2$ are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ touches $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \neq A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \neq B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.