Question 174
Let $ABC$ be a triangle with $AB = 2$, $CA = 3$, $BC = 4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $ABC$, and let $E$ lie on line $AD$ such that $D$ is the midpoint of $\overline{AE}$. Line $l$ passes through $E$ perpendicular to $\overline{AE}$, and $F$ and $G$ are the intersections of the extensions of $\overline{AB}$ and $\overline{AC}$ with $l$. Compute $FG$.