AMC 10 2014b Problem 15

In rectangle $ABCD$, $DC = 2 \cdot CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?

$\textbf{(A)}\ \ \frac{\sqrt{3}}{6} \textbf{(B)}\ \frac{\sqrt{6}}{8} \textbf{(C)}\ \frac{3\sqrt{3}}{16} \textbf{(D)}\ \frac{1}{3} \textbf{(E)}\ \frac{\sqrt{2}}{4}$