A short proof of the twelve-point theorem

We present a short elementary proof of the following twelve-point theorem. Let $M$ be a convex polygon with vertices at lattice points, containing a single lattice point in its interior. Denote by $m$ (respectively, $m^*$) the number of lattice points in the boundary of $M$ (respectively, in the boundary of the dual polygon). Then $m+m^*=12$.