Question 58
In an acute triangle $ABC$, $AB \gt AC$, the bisector of angle $BAC$ and side $BC$ intersect at point $D$, two points $E$ and $F$ are in sides $AB$ and $AC$, respectively, such that $B, C, F, E$ are concyclic. Prove that the circumcenter of triangle $DEF$ coincides with the innercenter of triangle $ABC$ if and only if $BE + CF = BC$.
