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We investigate the regular p-gonal prism tilings (mosaics) in the hyperbolic 3-space that were classified by I. Vermes in [12]and [13]. The optimal hyperball packings of these tilings are generated by the ``inscribed hyperspheres'' whose metric data can be calculated by our method -- based on the projective interpretation of the hyperbolic geometry -- by the volume formulas of J. Bolyai and R. Kellerhals, respectively. We summarize in some tables the data and the densities of the optimal hyperball packings to each prism tiling in the hyperbolic space H 3.

We present a definition for the co-punctuality of three lines in the hyperbolic plane in terms of line-orthogonality, correcting an omission in [2] and [1].

Positive Lω1ω definitions of point-inequality and noncollinearity in terms of collinearity, which are valid in plane hyperbolic geometry over arbitrary Archimedean ordered Euclidean fields, provide a synthetic proof of the theorem stated in the title and first noticed to be a corollary of a result from [2] by R. Hofer [3].

We present an axiom system for plane hyperbolic geometry in a language with lines as the only individual variables and the binary relation of line-perpendicularity as the only primitive notion. It was made possible by results obtained by K. List and H.L. Skala. A similar axiomatization is possible for n-dimensional hyperbolic geometry with n≥4. We also point out that plane hyperbolic geometry admits a AE-axiomatization in terms of line-perpendicularity alone, an axiomatization we could not find.

Packings of congruent circles have extensively been studied as well in the Euclidean plane as on the sphere and in the hyperbolic plane. In the present paper packings of more general domains are considered and the proof of a long claimed result of the author is given.