Casey Type Theorem and Laguerre Transformations

The article explores the connections between Casey's theorems and their generalizations on the Euclidean and pseudo-Euclidean planes. Along with Casey type theorems about circles and "tangent distances'' between them, Laguerre transformations that preserve such distances are considered. Using non-Euclidean geometry, some connections between such transformations are described. In Casey's theorem, which is one of the generalizations of Ptolemy's theorem on an inscribed quadrilateral, four circles are considered that are tangent to one circle on the Euclidean plane. Instead of the lengths of the sides and diagonals, Casey's theorem takes the lengths of the common tangents of the corresponding pairs of circles. This theorem can be easily generalized to a larger number of circles. In addition, this theorem has various analogs and generalizations in spaces of constant curvature. On the pseudo-Euclidean plane, one can also consider analogs of Casey's theorem and its generalizations. Theorems of this type on the pseudo-Euclidean plane are a direct consequence of the corresponding Euclidean theorems. In this paper, a correspondence is constructed between configurations of circles on the Euclidean plane and configurations of circles of imaginary radius on the pseudo-Euclidean plane. In this case, the relationship from Euclidean geometry corresponds to the same relationship in pseudo-Euclidean geometry. Laguerre transformations on the Euclidean plane affect oriented lines. In this case, the family of straight lines enveloping the circle, under the influence of Laguerre transformations, passes into a similar family. If a straight line belongs to two such families, then under Laguerre transformations the length of the straight line segment between the points of contact with the circles is preserved. Using isotropic projection, Laguerre transformations on Euclidean and pseudo-Euclidean planes can be considered as transformations induced by the movements of three-dimensional pseudo-Euclidean space. To describe the properties of one-parameter subgroups of the Laguerre group on the Euclidean and pseudo-Euclidean planes, the Lobachevsky and de Sitter geometries are used.