Here's a complicated-sounding geometry problem, Any help would be appreciated :)
Let $\triangle ABC$ be an obtuse-angled triangle with circumcentre $O$, circumcircle $\Gamma$ and $\angle ABC > 90^\circ$. Let $AB$ intersect the line through $C$ perpendicular to $AC$ in $D$. Let $l$ be the line through $D$ perpendicular to $AO$, and let $E$ be the intersection of $l$ and $AC$, and let $F$ be the point between $D$ and $E$ where $l$ intersects $\Gamma$. Can you prove that the circumcircles of triangles $\triangle BFE$ and $\triangle CFD$ are tangent at $F$?