BALTIC-WAY 1999 Problem 14

Let $A B C$ be an isosceles triangle with $|A B|=|A C|$. Points $D$ and $E$ lie on the sides $A B$ and $A C$, respectively. The line passing through $B$ and parallel to $A C$ meets the line $D E$ at $F$. The line passing through $C$ and parallel to $A B$ meets the line $D E$ at $G$. Prove that

$$\frac{[D B C G]}{[F B C E]}=\frac{|A D|}{|A E|}$$

where $[P Q R S]$ denotes the area of the quadrilateral $P Q R S$.