On generalizations of Ptolemy's theorem on the Lobachevsky plane

The article considers some variations on the theme of Ptolemy's theorem and its generalizations on the Lobachevsky plane. Some of the statements are hyperbolic analogues of the Euclidean theorems in terms of the figures involved in them. Other statements describe formal relations coinciding with Euclidean ones, but connecting other types of figures specific to the Lobachevsky plane. One of the generalizations is Fuhrmann's theorem — an analog of Ptolemy's theorem for an inscribed hexagon. If a hexagon on the Lobachevsky plane is inscribed in a circle, then the proof of the corresponding statement is not required: on the hyperbolic plane of curvature equal to minus one, it is obtained by standard substitution instead of the lengths of the segments of doubled hyperbolic sines by half their lengths. This article initially proves this statement for a hexagon inscribed in a horocycle and one branch of an equidistant line. In these cases, the sides and diagonals of the hexagon are related by the same relationship as for a hexagon inscribed in a circle. Then the cases are considered when from one to three vertices of the inscribed hexagon lie on the second branch of the equidistant line. In these cases, the analytical notation of the relations in Fuhrmann's theorem differs from the previous ones by replacing some hyperbolic sines of half the lengths of segments by hyperbolic cosines. In addition, analogues of Fuhrmann's theorem for six circles tangent to one fixed circle or horocycle are considered.