BALTIC-WAY 2006 Problem 13

In a triangle $A B C$, points $D, E$ lie on sides $A B, A C$ respectively. The lines $B E$ and $C D$ intersect at $F$. Prove that if

$$B C^{2}=B D \cdot B A+C E \cdot C A,$$

then the points $A, D, F, E$ lie on a circle.