Application of Renormalization Techniques to the Classical Arbelos Problem

TL;DR

This paper applies renormalization techniques to analyze the classical Arbelos problem, deriving formulas for semicircle areas in terms of a key chord and circumscribing circle radius, simplifying the problem's variables.

Contribution

It introduces a modern renormalization approach to the Arbelos problem, providing new formulas that reduce variable complexity compared to traditional methods.

Findings

Derived formulas for semicircle areas in terms of chord T and radius R

Eliminated dependence on additional variables of circumscribed semicircles

Provided a novel geometric analysis using renormalization techniques

Abstract

This application of renormalization techniques offers a modern take on the classical Arbelos geometry problem. Keeping within the context of the original problem, two semicircles, meeting at chord T, are together circumscribed by a third semicircle. Separate from the original Arbelos result, both circumscribed semicircle areas are found in terms of chord T and the third circumscribing semicircle radius R. This approach eliminates the additional variables of the circumscribed semicircle radii.