Question 97
Let $\triangle A B C$ be an isosceles triangle with $|A B|=|A C|$. A point $P$ inside $\triangle A B C$ is given, different from the centre of the circumcircle. Let $\omega$ be the circle through $C$ with centre $P$. It is given that the circle $\omega$ intersects the segments $B C$ and $A C$ a second time at $D$ and $E$, respectively. Let $\Gamma$ be the circumcircle of $\triangle A E P$ and let $F$ be the second intersection point of $\omega$ and $\Gamma$. Prove that the centre of the circumcircle of $\triangle B D F$ lies on $\Gamma$.
