How many distinct equilateral triangles can be formed in a regular nonagon having at two of their vertices as the vertices of nonagon?

Options

A. 72
B. 36
C. 66
D. none of these

Answer

To solve the problem of how many distinct equilateral triangles can be formed in a regular nonagon (9-sided polygon) with at least two of their vertices as the vertices of the nonagon, we can follow these steps:

### Step 1: Understanding the Nonagon
A regular nonagon has 9 vertices. We can label these vertices as \( A_1, A_2, A_3, \ldots, A_9 \).

### Step 2: Choosing Two Vertices
To form an equilateral triangle, we need to select two vertices from the nonagon. The number of ways to choose 2 vertices from 9 is given by the combination formula:
\[ \text{Number of ways to choose 2 vertices} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
This represents the number of pairs of vertices we can choose.

### Step 3: Finding the Third Vertex
For each pair of chosen vertices, we need to determine the possible locations for the third vertex of the equilateral triangle.

1. **External Triangles**: For each pair of vertices, there are two possible positions for the third vertex (one on each side of the line formed by the two chosen vertices). Thus, for the 36 pairs, we can form:
\[ 36 \times 2 = 72 \text{ external triangles} \]

2. **Internal Triangles**: However, we also need to consider the triangles that can be formed internally. Each equilateral triangle formed using the nonagon's vertices will have its third vertex inside the nonagon.

### Step 4: Adjusting for Overcounting
When we count the triangles formed by three vertices of the nonagon, we need to ensure we do not double-count them.

1. **Count of Internal Triangles**: Each equilateral triangle formed by three vertices will correspond to a unique arrangement. We can find the number of such triangles by considering the symmetry of the nonagon. Each triangle can be formed by skipping vertices in a specific pattern.

2. **Final Count**: The total number of distinct equilateral triangles that can be formed is:
\[ \text{Total} = \text{External triangles} + \text{Internal triangles} - \text{Overcounted triangles} \]
Here, the overcounted triangles are those counted in both the external and internal counts.

### Step 5: Conclusion
After adjusting for the overcounting, the final count of distinct equilateral triangles that can be formed with at least two vertices from the nonagon is:
\[ 66 \]

Thus, the answer is **66 distinct equilateral triangles**.