Olympiad Geometry · Superchallenge problem

Right triangle $ABC$ has its right angle at $A$, with $AB = 3$ and $AC = 4$. Square $BCDE$ is drawn on the hypotenuse, on the side of $\overline{BC}$ away from $A$. Segments $\overline{AD}$ and $\overline{AE}$ cross $\overline{BC}$ at $F$ and $G$. Find $FG$.