In rectangle A B C D, D C=2 C B and points E and F lie on \overline{A B} so that \overline{E D} and \overline{F D} trisect \angle A D C as shown. What is the ratio of the area of \triangle D E F to the area of rectangle A B C D ?

**Answer Choices**

A. \dfrac{\sqrt{3}}{6}

B. \dfrac{\sqrt{6}}{8}

C. \dfrac{3 \sqrt{3}}{16}

D. \dfrac{1}{3}

E. \dfrac{\sqrt{2}}{4}