How many isosceles triangles can be formed with 12 equally spaced points lying on the circumference of a circle?
The answer should be 52, but I have no idea how to solve it. Please help!
2026-03-30 02:24:08.1774837448
Number of isosceles triangles formed with 12 equally spaced points lying on the circumference of a circle
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How we're going to solve the question:
Pick a point, $A$, and label it the vertex opposite the base of the isosceles triangle. There are then $5$ ways to choose other points, hence since there are $12$ ways to choose $A$, there are $60$ ways to choose an isosceles triangle.
However, note that for equilateral triangles we overcount those by a factor of $3$. Thus we subtract twice the number of equilateral triangles (trivially $4$), as each gets counted two times too many, and we are left with $52$ as required.