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Tangents of incircles

Source: Costa Rica National Olympiad, Final Round, Problem 6

November 26, 2011
geometrygeometric transformationreflectionincenterparallelogramgeometry proposed

Problem Statement

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