On Analogues of the Fuhrmann's Theorem on the Lobachevsky Plane

Kostin, A. V.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 4.

Abstract:
According to Ptolemy's theorem, for a quadrilateral inscribed in a circle on the Euclidean plane, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of opposite sides. This theorem has various generalizations. On the plane in one of the generalizations an inscribed hexagon is considered instead of a quadrilateral. The corresponding statement relating the lengths of the sides and long diagonals of an inscribed hexagon is called Ptolemy's theorem for a hexagon or Fuhrmann's theorem. The Casey's theorem is another generalization of Ptolemy's theorem. In it, instead of four points lying on some fixed circle, four circles tangent to this circle are considered, whilst the lengths of the sides and diagonals are replaced by the lengths of the segments tangent to the circles are considered. If the curvature of Lobachevsky plane is equal to minus one, then in the analogues of the theorems of Ptolemy, Fuhrmann and Casey for polygons inscribed in a circle or circles tangent to one circles, the lengths of the corresponding segments, divided by two, will be under the signs of hyperbolic sines. In this paper, we prove theorems generalizing on the Lobachevsky plane Casey's theorem and Fuhrmann's theorem. On the Lobachevsky plane, six circles are considered that are tangent to some line of constant curvature, and for lengths tangent segments assertions generalizing these theorems are proved. 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 a correspondence can be established between the Euclidean and hyperbolic relations. This 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" the tangents will be connected by the corresponding "Euclidean" relation.

Keywords: Ptolemy's theorem, Casey's theorem, Fuhrmann's theorem, Lobachevsky plane, horocycle, equidistant

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