CZ-SK-MO 2003 Problem 2

On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$ . Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.