Polygons in Polygons with a Twist

TL;DR

This paper explores the construction of specific regular sub-n-gons within regular n-gons using special chords, revealing relationships involving areas that are divisors of the original polygon's area, aided by dynamic geometry software.

Contribution

It introduces a novel method for constructing regular sub-n-gons within regular n-gons using a special chord system, expanding understanding of polygon relationships.

Findings

Certain sub-n-gons have areas as divisors of the original polygon's area

Vertex-to-opposite-side chords facilitate specific sub-gon constructions

Dynamic Geometry software is essential for exploring these geometric relationships

Abstract

This is a study of the construction of particular regular sub-n-gons T in regular n-gons P using a special system of chords of P. In particular, some of these sub-n-gons have areas which are integer divisors of the area of the given n-gon P. Initially, the study will concentrate on chords which are from a vertex to special points of one of the opposite sides of P. Several examples are explored. However, it will become apparent that a much more general situation exists. Dynamic Geometry software is the key to investigating this new relationship.