Two particles move along the edges of equilateral $\triangle ABC$ in the direction\[A\Rightarrow B\Rightarrow C\Rightarrow A,\]starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$? $\mathrm {(A)} \frac{1}{16}\qquad$ $ \mathrm {(B)} \frac{1}{12}\qquad$ $ \mathrm {(C)} \frac{1}{9}\qquad$ $ \mathrm {(D)} \frac{1}{6}\qquad$ $ \mathrm {(E)} \frac{1}{4}$