Question 22
Let $ABC$ be an isosceles triangle with $|AB| = |AC|$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that
$$
\frac{[DBCG]}{[FBCE]} = \frac{|AD|}{|AE|}
$$
where $[PQRS]$ denotes the area of the quadrilateral $PQRS$.
$$
\frac{[DBCG]}{[FBCE]} = \frac{|AD|}{|AE|}
$$
where $[PQRS]$ denotes the area of the quadrilateral $PQRS$.