Apollonius surfaces, circumscribed spheres of tetrahedra, Menelaus' and Ceva's theorems in $\SXR$ and $\HXR$ geometries

TL;DR

This paper explores generalized Apollonius surfaces, circumscribed spheres of tetrahedra, and classical theorems in $ ext{S}^2 imes ext{R}$ and $ ext{H}^2 imes ext{R}$ geometries, extending geometric concepts to Thurston 3-geometries.

Contribution

It introduces generalized Apollonius surfaces and extends Menelaus' and Ceva's theorems to $ ext{S}^2 imes ext{R}$ and $ ext{H}^2 imes ext{R}$ geometries, providing new tools for geometric analysis.

Findings

Defined and characterized generalized Apollonius surfaces in $ ext{S}^2 imes ext{R}$ and $ ext{H}^2 imes ext{R}$.

Developed a method to find circumscribed spheres of tetrahedra in these geometries.

Generalized classical theorems for geodesic triangles in the studied geometries.

Abstract

In the present paper we study $\SXR$ and $\HXR$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the "surface of a geodesic triangle". Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\SXR$ and $\HXR$ tetrahedron. Moreover, we generalize the famous Menelaus' and Ceva's theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\SXR$ and $\HXR$ geometries described by E. Moln\'ar in \cite{M97}.