Question 55
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$ such that $|AC| \neq |BC|$. The internal angle bisector of $\angle CAB$ intersects side $[BC]$ in $D$, and the external angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $\Omega$ again in $E$ and $F$, respectively. Let $G$ be the intersection of lines $AE$ and $FI$ and let $\Gamma$ be the circumcircle of triangle $BDI$. Show that $E$ lies on $\Gamma$ if and only if $G$ lies on $\Gamma$.
