A short proof of the Twelve points theorem

TL;DR

This paper provides a concise, elementary proof of the Twelve Points Theorem, which relates lattice points on a convex polygon and its dual, emphasizing simplicity and clarity in the proof process.

Contribution

It introduces a short, elementary proof of the Twelve Points Theorem, simplifying previous approaches and making the result more accessible.

Findings

Proves the Twelve Points Theorem using an elementary approach

Establishes a clear relationship between lattice points of a polygon and its dual

Simplifies understanding of lattice point enumeration in convex polygons

Abstract

We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m*) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then m+m*=12.