In rectangle $ABCD$, $AB=2$, $BC=6$, and points $E,F,G$ are midpoints of $\overline{BC},\overline{CD},\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?

1. The diagonals and midpoint segments give $K=(\frac43,2)$ and $K'=(\frac23,2)$, while $H=(1,3)$ and $F=(1,0)$.
2. The shaded quadrilateral $KHK'F$ has diagonals $KK'=\frac23$ and $HF=3$, so its area is $\frac12\cdot\frac23\cdot3=\boxed{1}$.