CMO 2014 Problem 1

Let $ABC$ be a triangle with $AB>AC$ . Let $D$ be the foot of the internal angle bisector of $A$ . Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$ .