# Difference between revisions of "2001 AMC 12 Problems/Problem 14"

## Problem

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 $\{A_1,A_2,\dots,A_9\}$? $\text{(A) }30 \qquad \text{(B) }36 \qquad \text{(C) }63 \qquad \text{(D) }66 \qquad \text{(E) }72$

## Solution

Each of the ${9\choose 2}=36$ pairs of vertices determines two equilateral triangles, one on each side of the segment. This would give us $72$ triangles. However, note that there are three equilateral triangles that have all three vertices among the vertices of the polygon. These are the triangles $A_1A_4A_7$, $A_2A_5A_8$, and $A_3A_6A_9$. We counted each of these three times (once for each side). Hence we overcounted by $2$ for each of these triangles for a total of $6$ overcounted, and the correct number of equilateral triangles is $72-6=\boxed{66}$.

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