General method of Complex Polynomial for determining the radius of the circle circumscribed to a cyclic polygon an arbitrary number of sides, and some important consequences

TL;DR

This paper introduces a general complex number-based method to determine the circumscribed circle radius of cyclic polygons with any number of sides, enabling area calculations and exploring properties of non-convex cyclic polygons.

Contribution

It presents a novel, general approach using complex numbers to find the radius of the circumscribed circle for cyclic polygons with arbitrary sides, extending existing methods.

Findings

Method applicable to polygons with any number of sides

Allows calculation of polygon area from side lengths and radius

Provides insights into properties of non-convex cyclic polygons

Abstract

This paper presents a general method for obtaining radius of the corresponding circumference to a cyclical polygon $n$ sides given the lengths of said sides, using the notion of complex number. As of radius $r$, obtained, can then be calculated polygon area in question applying known expressions which requires, after $n=4$, a powerful calculation tool. Are also given like elements regarding non convex polygons cyclic, although in this respect it deepens less and is not considered, course, the calculation of areas.