On a Family of Polygons

Abstract

Because polygons are easy to define, it is sometimes thought that they only possess trivial properties. However, sometimes even quite obvious properties turn out to be tricky to establish. Let vI, v2,...,v,, be n given vectors in the plane, having different directions and satisfying El Ivi = 0. Each vector can be translated in the plane without affecting its length or direction, and so by performing suitable translations, a polygon having the n vectors as edges is produced. Such a polygon may be convex, nonconvex, or self-intersecting (see FIGURES 1 and 2). There are (n 1)! distinct polygons that can be formed in this way (two polygons are equivalent if one can be obtained from the other by translation). We denote by S,, the set of distinct polygons having the n vectors vi as edges. If P, is a polygon of S,, we define the area of P,, A (P,,), to be the area of the finite region of the plane bounded by P,. We establish two properties of S,,: