CR-MO 2011 Problem 6

Let $ABC$ be a triangle. The incircle of $ABC$ touches $BC,AC,AB$ at $D,E,F$ , respectively. Each pair of the incircles of triangles $AEF, BDF,CED$ has two pair of common external tangents, one of them being one of the sides of $ABC$ . Show that the other three tangents divide triangle $DEF$ into three triangles and three parallelograms.