AHSME 1993 Problem 25

Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are $\text{(A) exactly 2 such triangles} \\ \text{(B) exactly 3 such triangles} \\ \text{(C) exactly 7 such triangles} \\ \text{(D) exactly 15 such triangles} \\ \text{(E) more than 15 such triangles}$