We define and determine the generalized Apollonius surfaces and with
them define the ``surface of a geodesic triangle" in the above geometries.

Using the above Apollonius surfaces we develop a procedure to determine the centre
and the radius of the circumscribed geodesic sphere of an arbitrary S^2xR and H^2 xR
tetrahedron.

We generalize the famous Menelaus' and Ceva's theorems for geodesic triangles
in the S^2xR and H^2 xR spaces and show that in the Nil space the ``lines" on the surface of a geodesic triangle can be defined by the famous Menelaus' condition and prove
that Ceva's theorem for geodesic triangles is true in Nil space.