A remark on Chapple-Euler Theorem in non Euclidean geometry

TL;DR

This paper discusses variations of the Chapple-Euler Theorem in non-Euclidean geometry, demonstrating that these results lead to expressions similar to the classical formula involving the circumradius and inradius.

Contribution

It shows that known non-Euclidean versions of the Chapple-Euler Theorem produce formulas analogous to the Euclidean case, connecting different geometric contexts.

Findings

Non-Euclidean versions yield formulas similar to $d=\,\sqrt{R(R-2r)}$

Results unify Euclidean and non-Euclidean geometric relations

Provides insight into the structure of non-Euclidean geometry

Abstract

In non-Euclidean geometry, there are several known correspondings to Chapple-Euler Theorem. This remark shows that those results yield expressions corredponding to the well-known formula $d=\sqrt{R(R-2r)}$.