Apollonius曲面，四面体的外切球面，$\SXR$和$\HXR$几何中的Menelaus和Ceva定理
Apollonius surfaces, circumscribed spheres of tetrahedra, Menelaus' and Ceva's theorems in $\SXR$ and $\HXR$ geometries
摘要：
本文研究了齐次Thurston 3-几何$\SXR$和$\HXR$.我们定义和确定广义Apollonius曲面，并与他们定义的“曲面的测地线三角形”。利用上述Apollonius曲面，我们发展了一个确定任意$\SXR$和$\HXR$四面体的外接测地球的中心和半径的方法.此外，我们推广了著名的Menelaus和Ceva定理的测地三角形在这两个空间。在我们的工作中，我们将使用由E.在\cite{M97}的Molnár。
英文摘要：
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}.
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