  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}.
