Apollonius surfaces, circumscribed spheres of tetrahedra, Menelaus' and Ceva's theorems in $\SXR$ and $\HXR$ geometries
arXiv:2012.06155
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. Molnár in \cite{M97}.
23 pages, 10 figures