Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AF}$, and let $H$ be the midpoint of $\overline{CD}$. What is the perimeter of $GCHF$?

1. Use coordinates $A=(0,2)$, $B=(\sqrt3,1)$, $C=(\sqrt3,-1)$, $D=(0,-2)$, $E=(-\sqrt3,-1)$, and $F=(-\sqrt3,1)$. The requested midpoints are $G=(-\sqrt3/2,3/2)$ and $H=(\sqrt3/2,-3/2)$.
2. The distance formula gives $GC=HF=\sqrt{(3\sqrt3/2)^2+(5/2)^2}=\sqrt{13}$ and $CH=FG=\sqrt{(\sqrt3/2)^2+(1/2)^2}=1$.
3. Therefore, the perimeter of $GCHF$ is $2\sqrt{13}+2$.