### Introduction

Usually, a polygon is considered to be a finite portion of the plane limited exclusively by line segments (sides) that intersect only at its ends (vertices), each vertex being common to exactly two sides. However, a polygon may also be considered as a finite sequence \((P_0, P_1, P_2, \ldots, P_{n-1})\) of not necessarily distinct points in the plane (or, more generally, in space). These points are the vertices of the polygon. To better visualize this sequence, one usually draws line segments joining the consecutive vertices, that is, joining the vertices \(P_i\), \(P_{i+1}\), with \(i\in\{0,1,\ldots,n-1\}\), where \(P_0= P_n\). Thus, we consider as polygons the following plane shapes:

We assign to each vertex \(P_i\) its coordinates, given by an ordered pair \((x_i,y_i\)) or an ordered triple \((x_i,y_i,z_i\)), depending on whether we are considering points on the plane or in space. Let \(P'_i\) denote the midpoint of the line segment linking vertices \(P_i\) and \(P_{i+1}\). Its coordinates can be obtained from the coordinates of \(P_i\) and \(P_{i+1}\). We also assign to \(P'_i\) its pair \((x'_i,y'_i\)) or triple which are given by the identities \[x'_{i'}=\frac{x_i+x_{i+1}}{2}\] \[y'_{i'}=\frac{y_i+y_{i+1}}{2}\] \[z'_{i'}=\frac{z_i+z_{i+1}}{2}\] with \(i\in\{0,1,\ldots,n-1\}\).

Note that all coordinates of \(P'_i\) are obtained from the corresponding coordinates of \(P_i\) and \(P_{i+1}\) in the same way (by their arithmetic mean). We will only consider transformations where this happens, so we only need to know how to obtain the abscissa of the new points from the abscissa of the points of the original polygon, since the other coordinates are obtained in a similar way. Therefore, we will not need to distinguish between points on the plane or space, and all valid constructions on the plane (such as e.g. a bisection) are also valid in space.