Question 88
Let $A$, $B$, $C$ be points lying on a circle $\Gamma$ with the center $O$ and assume that $\angle ABC \gt 90^\circ$. Let $D$ be the point of intersection of the line $AB$ and the line perpendicular to $AC$ at $C$. Let $l$ be the line through $D$ and perpendicular to $AO$. Let $E$ be the point of intersection of $l$ and the line $AC$, and $F$ be the point of intersection of $\Gamma$ and $l$ that lies between $D$ and $E$. Prove that the circumcircles of the triangles $BFE$ and $CFD$ are tangent at $F$.