2000 Gauss 8 Problem 24

In equilateral triangle ABC, line segments are drawn from a point P to the vertices A, B and C to form three identical triangles. The points D, E and F are the midpoints of the three sides and they are joined as shown in the diagram. What fraction of \triangle ABC is shaded?

\textbf{(A)}\ \frac{1}{5}\quad \textbf{(B)}\ \frac{5}{24}\quad \textbf{(C)}\ \frac{1}{4}\quad \textbf{(D)}\ \frac{2}{9}\quad \textbf{(E)}\ \frac{2}{7}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.