After drawing the diagram, we can easily see that $AD$, $BE$, and $CF$ are the symmedians of $DEF$ and $GHI$ so they concur at their Lemoine point. This means that $X$ is the reflection of the midpoint of $HI$ over the $G$-angle bisector of $\triangle GHI$. After some simple computing, we find that $IX=\frac{7}{2}$ so our answer is $\boxed{9}$.

kn07 wrote: After drawing the diagram, we can easily see that $AD$, $BE$, and $CF$ are the symmedians of $DEF$ and $GHI$ so they concur at their Lemoine point. This means that $X$ is the reflection of the midpoint of $HI$ over the $G$-angle bisector of $\triangle GHI$. After some simple computing, we find that $IX=\frac{7}{2}$ so our answer is $\boxed{9}$. Mmm, that's not right. Recheck your calculations.