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.

  • $\begingroup$ It is easy to draw a regular n-gon in GeoGebra with its centre marked. Experiment with the triangles you can form. $\endgroup$
    – Paul
    Feb 11 '15 at 14:16
  • $\begingroup$ Have you seen my answer, Pankaj? $\endgroup$ Feb 17 '15 at 11:19
  • $\begingroup$ Yes . Answer is correct. But the link you have mentioned does not tell how to find the required number of ways. $\endgroup$
    – Maverick
    Feb 17 '15 at 14:00

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 https://oeis.org/A000330

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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