# Find the number of triangles formed by the vertices of a polygon of $2n+1$ sides each containing the centre of the polygon.

Find the number of triangles formed by the vertices of a polygon of $2n+1$ sides each containing the centre of the polygon.

I found that for $2k+1$ sides it is coming $1^2+2^2+3^2+..k^2$.I wanted to try induction for $2k+3$, but I think there is some undercounting..

Please help

• The phrasing of this problem is horrible...does "each containing the center" refer to the edges, or does it narrow the set of polygons we care about? Also, assuming that the polygon is a regular $n$-gon, your answer for $n = 3$ is $1 + 4 + 9 = 14$, but the correct answer seems to be 1. Probably best not to try an inductive proof until you have a conjecture that works in the base case. – John Hughes Apr 19 '16 at 3:12
• Also: if you take a perfect square, is the answer 0 or 8? In other words, if the polygon center lies on an edge of the triangle, does the triangle contain the center or not? – John Hughes Apr 19 '16 at 3:14
• Yes,I meant each such triangle contains the center – Legend Killer Apr 19 '16 at 3:24
• Do you mean a regular polygon? If so, induction will be tough. If not, then what do you mean by "center"? And does the polygon have to be non-self-intersecting, i.e., be a simple closed curve in the plane? And is the center "in" a triangle if it's on one of the edge, or must it be in the interior of the triangle? – John Hughes Apr 19 '16 at 12:16