IrMO 2009 Problem 10

In the triangle $ABC$ we have $|AB|<|AC|$ . The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$ .

Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$