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In the present paper we deal with non-constant curvature Thurston geometries \cite{M97}, \cite{S}, \cite{Sz22-3},\cite{W06}. We define and determine the generalized trans\-lation-like Apollonius surfaces and thus also bisector surfaces as a special case. Moreover, we give a possible definition of the "surface of a translation-like triangle" in each investigated geometry. In our work we will use the projective model of Thurston geometries described by E. Moln\'ar in \cite{M97}., Comment: arXiv admin note: text overlap with arXiv:1705.04207

After having investigated the geodesic triangles and their angle sums in Nil and $Sl\times\mathbb{R}$ geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to $\pi$. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work we will use the projective model of Sol described by E. Moln\'ar in \cite{M97}.

In this work, we examine the isoptic surface of line segments in the $S^2\times R$ and $H^2\times R$ geometries, which are from the 8 Thurston geometries. Based on the procedure first described in [10], we are able to give the isoptic surface of any segment implicitly. We rely heavily on the calculations published in [41, 42]. As a special case, we examine the Thales sphere in both geometries, which are called Thaloid. In our work we will use the projective model of $S^2\times R$ and $H^2\times R$ described by E. Moln\'ar in [20].

After having investigated the geodesic and translation triangles and their angle sums in $\SOL$ and $\SLR$ geometries we consider the analogous problem in $\NIL$ space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in $\NIL$ geometry and we provide a new approach to prove that it can be larger than or equal to $\pi$. Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of $\NIL$ isoptic surfaces of translation-like segments and as a special case of this we examine the $\NIL$ translation-like Thales sphere, which we call {\it Thaloid}. In our work we will use the projective model of $\NIL$ described by E. Moln\'ar in \cite{M97}., Comment: 22 pages, 6 figures. arXiv admin note: text overlap with arXiv:1710.02394, arXiv:1703.06646

In this paper we deal with $\NIL$ geometry, which is one of the homogeneous Thurston 3-geometries. We define the "surface of a geodesic triangle" using generalized Apollonius surfaces. Moreover, we show that 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. In our work we will use the projective model of $\NIL$ geometry described by E. Moln\'ar in \cite{M97}., Comment: 19 pages, 3 figures. arXiv admin note: text overlap with arXiv:2012.06155

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}., Comment: 23 pages, 10 figures

In the present paper we study $S^2 \times R$ and $H^2 \times R$ geometries, which are homogeneous Thurston 3-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and prove, that in $S^2 \times R$ space it can be larger or equal than $\pi$ and in $H^2 \times R$ space the angle sums can be less or equal than $\pi$. In our work we will use the projective model of $S^2 \times R$ and $H^2 \times R$ geometries described by E. Moln\'ar in \cite{M97}., Comment: 22 pages and 10 figures. arXiv admin note: substantial text overlap with arXiv:1703.06646, arXiv:1611.05613

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}., Comment: 17 pages, 8 figures. arXiv admin note: text overlap with arXiv:1703.06646

After having investigated the geodesic and translation triangles and their angle sums in $\NIL$ and $\SLR$ geometries we consider the analogous problem in $\SOL$ space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of translation triangles in $\SOL$ geometry and prove that it can be larger or equal than $\pi$. In our work we will use the projective model of $\SOL$ described by E. Moln\'ar in \cite{M97}, Comment: 13 pages, 4 figures

In this paper we study the interior angle sums of geodesic triangles in $\NIL$ geometry and prove that it can be larger, equal or less than $\pi$. We use for the computations the projective model of $\NIL$ introduced by E. {Moln\'ar} in \cite{M97}., Comment: 15 pages, 4 figures. arXiv admin note: text overlap with arXiv:1607.04401, arXiv:1610.01500