Olympiad Geometry · Challenge problem

In $\triangle ABC$, medians $\overline{AD}$ and $\overline{BE}$ (with $D$ the midpoint of $\overline{BC}$ and $E$ the midpoint of $\overline{AC}$) intersect at the centroid $G$. If $AG = 12$, $BG = 8$, and $m\angle AGB = 90^\circ$, find $DE$.