The answers to a recent question established that it is possible to construct families of polygons all with the same area and perimeter. Some comments on some of the answers inspired this very specific question:
Prove that for any n-sided polygon P, and any integer m greater than n, there is an m-sided polygon with the same area and perimeter as P.
- I define a polygon as not having two successive edges collinear, so you can't just insert a vertex to the middle of an edge.
- I do not care if the polygons in question are convex or not. So it needs to work if P is not convex, but it does not need to produce convex polygons.
- I would like a proper written proof, rather than just a description of how one might construct a proof.