In a circle $C(O(0,0),1)$ with a polygon inscribed $A_1A_2...A_n$, where $n \in \mathbb{N}, n\ge3$, such that $O$ is situated in the interior of the polygon. Let $G$ be the centroid (mass center) of the polygon. Prove that $OG \le \frac{n-2}{n}$ and find all the values of $n$ for the equality.
Please, someone can help me? I have no clue. I know that there is a formula for the centroid veriticies but i don't know how this can help me.I found this formula for the centroid of a polygon in terms of its coordinates.
$C_x =\frac{1}{6A} \sum_{i=0}^{N-1}(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}y_i) $
$C_y =\frac{1}{6A} \sum_{i=0}^{N-1}(y_i+y_{i+1})(x_iy_{i+1}-x_{i+1}y_i)$
where $x_N=x_0$ and A = the area of the polygon.
Obviously, my polygon is convex and cyclic because it's inscribed in the circle.
Any ideas?