  In the present paper we study the $\SOL$ 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.
  Moreover, we develop a method to determine the centre and the radius of the
circumscribed translation sphere of a given {\it translation tetrahedron}.
  In our work we will use for computations and visualizations the projective
model of $\SOL$ described by E. Moln\'ar in \cite{M97}.
