Question 191
Two circles $\Gamma_1$ and $\Gamma_2$ intersect at two distinct points $A$ and $B$. A line $\ell$ through $A$ meets $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$ respectively, such that $D$ lies inside $\Gamma_1$. The line perpendicular to $\ell$ through $A$ meets $\Gamma_1$ and $\Gamma_2$ at points $E$ and $F$ respectively. The line $EC$ meets the line $FD$ (extended) at the point $X$. Prove that if $X$ lies on the perpendicular bisector of $EF$, then $BX$ bisects the angle $\angle CBD$.