On Integer Geometry

In many questions, the geometric approach gives an intuitive visualization that leads to a better understanding of a problem and sometimes even to its solution. This chapter is entirely dedicated to notions, definitions, and basic properties of integer geometry. We start with general definitions of integer geometry, and in particular, define integer lengths, distances, areas of triangles, and indexes of angles. Further we extend the notion of integer area to the case of arbitrary polygons whose vertices have integer coordinates. Then we formulate and prove the famous Pick’s formula that shows how to find areas of polytopes simply by counting points with integer coordinates contained in them. Finally we formulate one theorem in the spirit of Pick’s theorem: it is the so-called twelve-point theorem.