The Descartes circles theorem and division by zero calculus

TL;DR

This paper explores the application of division by zero calculus to Descartes' circle theorem, extending its validity to lines and points and deriving new results for exceptional cases.

Contribution

It introduces a division by zero calculus framework and applies it to extend Descartes' theorem to new geometric configurations.

Findings

The theorem is valid for lines and points with one exception.

Division by zero calculus provides interesting results in exceptional cases.

New geometric insights are obtained through this extended framework.

Abstract

From the viewpoint of the division by zero $(0/0=1/0=z/0=0)$ and the division by zero calculus, we will show that in the very beautiful theorem by Descartes on three touching circles is valid for lines and points for circles except for one case. However, for the exceptional case, we can obtain interesting results from the division by zero calculus.