Question 27

$$A_1,A_2,A_3,\ldots,A_{15}$$ is a $$15$$ sided regular polygon. The number of distinct equilateral triangles in the plane of the polygon, with exactly two of their vertices from the set $$\{A_1,A_2,A_3,\ldots,A_{15}\}$$ is

Correct Answer: 195

Solution

Each pair of the $$15$$ polygon vertices determines two possible equilateral triangles in the plane, giving $$2\binom{15}{2}=210$$. However, the $$15$$ equilateral triangles formed entirely by polygon vertices have been counted, so they must be removed. Therefore the required number is $$210-15=195$$.

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$$A_1,A_2,A_3,\ldots,A_{15}$$ is a $$15$$ sided regular polygon. The number of distinct equilateral triangles in the plane of the polygon, with exactly two of their vertices from the set $$\{A_1,A_2,A_3,\ldots,A_{15}\}$$ is

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