IrMO 2006 Problem 7

$ABC$ is a triangle with points $D$ , $E$ on $BC$ with $D$ nearer $B$ ; $F$ , $G$ on $AC$ , with $F$ nearer $C$ ; $H$ , $K$ on $AB$ , with $H$ nearer $A$ . Suppose that $AH=AG=1$ , $BK=BD=2$ , $CE=CF=4$ , $\angle B=60^\circ$ and that $D$ , $E$ , $F$ , $G$ , $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$ .