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.