The number of perpendicularly inscribed polygons that intersect a given side in an odd sided regular polygon

TL;DR

This paper investigates the count of perpendicular inscribed polygons intersecting a specific side of odd-sided regular polygons, employing combinatorial, matrix, and fixed point methods to derive formulas and solutions.

Contribution

It introduces a novel approach combining circular permutations, circulant matrices, and Banach's Fixed Point Theorem to analyze inscribed polygons in regular polygons.

Findings

Derived formulas for counting such polygons.

Developed a method based on Banach's Fixed Point Theorem.

Calculated special cases using circulant matrices and the Binomial Theorem.

Abstract

The goal of this paper is to determine the number of perpendicularly inscribed polygons that intersect a given side of a regular polygon with an odd number of sides. This is done using circular permutations with repetition, and some special cases are calculated via circulant matrices and the Binomial Theorem. A method for finding such polygons, based on Banach's Fixed Point Theorem, is also developed.