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Consider a $2n+1$ sided regular polygon.In how many ways can we choose $3$ vertices out of these $2n+1$ vertices so that the centre of the polygon always lies inside the triangle formed by joining these $3$ chosen vertices.

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1 Answer

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I think this is the same as the number of acute triangles made from the vertices of a regular $(2n+1)$-polygon, which is $n(n+1)(2n+1)/6$, as given at

That is, I think you can prove that a triangle contains the center if and only if it is acute. The proof uses the theorem that the angle subtended by an arc at the center is twice the angle subtended by that arc from a point on the circumference.

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