How to find the number of acute angle and obtuse angled triangles that can be inscribed in a circle containing 'N' equally spaced points.
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2To clarify: You are given $N$ equally spaced points on the circle, and each vertex of the inscribed triangles must be one of those points, correct? What have you tried? – epimorphic Apr 21 '15 at 16:27
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Hint The total number of triangles that can be formed from $N$ points on the circle will be $\binom{N}{3}$. I'm assuming that you are looking for total number of acute and obtuse angled triangles. If that is the case then from this total number found above, subtract the ones which are right angled. For right angled think in terms of two points that lie on the diameter of a circle and the third on the circumference. Parity of $N$ will play some role.
If you are looking for separate count of acute and obtuse triangles then also think in terms of all 3 points lying on one side of a diameter as opposed two lying on one side and the third on the other.
Anurag A
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Hint: number the points from $1$ through $N$ around the circle. Consider a triangle that includes point $1$ and two other points. What can you say about the pairs of points that produce a right angle at point $1$? What can you say about the pairs of points that produce an obtuse angle at point $1$?
Ross Millikan
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