BALTIC-WAY 1998 Problem 14

Given a triangle $A B C$ with $|A B|<|A C|$. The line passing through $B$ and parallel to $A C$ meets the external bisector of angle $B A C$ at $D$. The line passing through $C$ and parallel to $A B$ meets this bisector at $E$. Point $F$ lies on the side $A C$ and satisfies the equality $|F C|=|A B|$. Prove that $|D F|=|F E|$.