On Analogs of Fuhrmann’s Theorem on the Lobachevsky Plane

Abstract

According to Ptolemy’s theorem, the product of the lengths of the diagonals of a quadrilateral inscribed in a circle on the Euclidean plane equals the sum of the products of the lengths of opposite sides. This theorem has various generalizations. In one of the generalizations on the plane, a quadrilateral is replaced with an inscribed hexagon. In this event the lengths of the sides and long diagonals of an inscribed hexagon is called Ptolemy’s theorem for a hexagon or Fuhrmann’s theorem. Casey’s theorem is another generalization of Ptolemy’s theorem. Four circles tangent to this circle appear instead of four points lying on some fixed circle whilst the lengths of the sides and diagonals are replaced by the lengths of the segments tangent to the circles. If the curvature of the Lobachevsky plane is \( -1 \), then in the analogs of the theorems of Ptolemy, Fuhrmann and Casey for the polygons inscribed in a circle or circles tangent to one circle, the lengths of the corresponding segments, divided by 2, will be under the signs of hyperbolic sines. In this paper, we prove some theorems that generalize Casey’s theorem and Fuhrmann’s theorem on the Lobachevsky plane. The theorems involve six circles tangent to some line of constant curvature. We prove the assertions that generalize these theorems for the lengths of tangent segments. If, in addition to the lengths of the segments of the geodesic tangents, we consider the lengths of the arcs of the tangent horocycles, then there is a correspondence between the Euclidean and hyperbolic relations, which can be most clearly demonstrated if we take a set of horocycles tangent to one line of constant curvature on the Lobachevsky plane. In this case, if the length of the segment of the geodesic tangent to the horocycles is \( t \), then the length of the “horocyclic” tangent to them is equal to \( \sinh\frac{t}{2} \). Hence, if the geodesic tangents are connected by a “hyperbolic” relation, then the “horocyclic” tangents will be connected by the corresponding “Euclidean” relation.