CEMC Cayley, 2002

In the diagram, triangle $ABC$ is isosceles with $AB = AC$, and $AG$ is perpendicular to $BC$. Point $D$ is the midpoint of $AB$, point $F$ is the midpoint of $AC$, and $E$ is the point of intersection of $DF$ and $AG$. What fraction of the area of $\triangle ABC$ does the shaded area represent?

(A) $\frac{1}{12}$ (B) $\frac{1}{6}$ (C) $\frac{1}{4}$ (D) $\frac{1}{10}$ (E) $\frac{1}{8}$