Question 67
In isosceles triangle $ABC$ with $AB = AC$, point $O$ is in its interior (not including circumference) and the circle $\omega$ centered $O$ and passing through $C$ intersects the sides (excluding end points) $BC$ and $AC$ at $D$ and $E$, respectively. Let $\Gamma$ be the circumcircle of triangle $AEO$ and intersect with $\omega$ again at $F \neq E$. Prove that the circumcenter of the triangle $BDF$ lies on $\Gamma$. In the above, denote by $XY$ the length of line segment $XY$.