Three distinct vertices are choose at random from the vertices of a given regular polygon of $2n+1$ sides. Each of those three vertices will determine a triangle. What is the probability that the center of the polygon lies inside the triangle determined by three vertices of the polygon?

Note: All vertices of a regular polygon lie on a common circle (the circumscribed circle), the center of this circle is the center of the polygon.

P.S: why is this question asked specially for a odd sided regular polygon, will the answer differ for a even sided regular polygon?

I have just found a difficult proof of a Much more general question (Probability of a fixed point in a convex region being inside a triangle formed by any three point form the convex region). I am not interested in such generality, can one give a much simpler proof this special case (the convex region being the odd-sided polygon and the fixed point the center of the polygon).