2001 AMC 12 Problem 14

Given the nine-sided regular polygon A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set \{A_1,A_2,\ldots,A_9\}?

\text{(A) }30 \qquad \text{(B) }36 \qquad \text{(C) }63 \qquad \text{(D) }66 \qquad \text{(E) }72

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.