Question 2
Equilateral triangle $ABC$ has side length $840.$ Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}.$ The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F,$ respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G,$ $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8 : 9.$ Find $AF.$
