RO-TST 2005 Problem 8

Let $ABC$ be a triangle, and let $D$ , $E$ , $F$ be 3 points on the sides $BC$ , $CA$ and $AB$ respectively, such that the inradii of the triangles $AEF$ , $BDF$ and $CDE$ are equal with half of the inradius of the triangle $ABC$ . Prove that $D$ , $E$ , $F$ are the midpoints of the sides of the triangle $ABC$ .