BdMO Online Forum

Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.

By angle chasing,
$\angle EAC=\angle B= \angle EAD$
So $ACED$ is cyclic and by radical axis theorem on $(ACBF),(EDBF),(ACED)$ we are done.