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Twelve points are spaced evenly around a circle, lettered from A to L. Let N be the total number of isosceles triangles, including equilateral triangles, that can be constructed from three of these points. A different orientation of the same lengths counts as a different triangle, because a different combination of points form the vertices. What is the value of N?

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sqdancefan

Answer:

  N = 52

Step-by-step explanation:

We assume that a 1-5-9 equilateral triangle is indistinguishable from a 5-9-1 equilateral triangle, so only count it once.

There are 5 isosceles triangles that can be created using each lettered vertex as the apex. Of those, 1 is equilateral, so there are 4×12 non-equilateral isosceles triangles that can be formed.

There are 4 unique equilateral triangles, so the total number is ...

  N = 4×12 + 4 = 52

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