  In this paper we study the Nil geometry that is one of the eight homogeneous
Thurston 3-geomet\-ri\-es.
  We determine the equation of the translation-like bisector surface of any two
points. We prove, that the isosceles property of a translation triangle is not
equivalent to two angles of the triangle being equal and that the triangle
inequalities do not remain valid for translation triangles in general. We
develop a method to determine the centre and the radius of the circumscribed
translation sphere of a given {\it translation tetrahedron}.
  A further aim of this paper is to study lattice-like coverings with congruent
translation balls in Nil space. We introduce the notion of the density of the
considered coverings and give upper estimate to it using the radius amd the
volume of the circumscribed translation sphere of a given {\it translation
tetrahedron}. The found minimal upper bound density of the translation ball
coverings $\Delta \approx 1.42783$. In our work we will use for computations
and visualizations the projective model of Nil described by E. Moln\'ar in
\cite{M97}.

Metric Geometry