May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a convex polygon in the plane and $v_{m+1}$ be a vertex in the interior of the convex polygon. Connect all the vertices by edges, and let $\alpha_{m}$ be the smallest angle among all the angles formed by two edges coming from the same vertex. Is it true that $m^{2}\alpha_{m}$ is bounded by an absolute constant (independent of $m$ and the $v$'s)? Any helpful answers would be greatly appreciated.

Let $m$ be odd, $v_1, \ldots, v_m$ be the vertices of a regular $m$gon, and $v_{m+1}$ be its centre. The smallest angle only involving vertices of the $m$gon is $\pi/m$, the angle over any edge when viewed from another vertex. The smallest angle obtained using the centre is half of this, $\pi/2m$. So $m^2\alpha_m$ is not bounded. 

