Five circles theorem

Abstract: The paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.

Abstract: The Circle in Elementary Plane Geometry: Fundamental definitions and notation Inversion Mutually tangent circles Circles related to a triangle The Brocard figures Concurrent circles and concyclic points Coaxal circles The Circle in Cartesian Plane Geometry: The circle studied by means of trilinear coordinates Fundamental relations, special tetracyclic coordinates The identity of Darboux and Frobenius Analytic systems of circles Famous Problems in Construction: Lemoine's geometrographic criteria Problem of Apollonius, number of real solutions Construction of Apollonius Construction of Gergonne Steiner's problem Circle meeting four others at equal or supplementary angles Malfatti's problem, Hart's proof of Steiner's construction Analytic solution, extension to thirty-two cases Examples of Fiedler's general cyclographic methods Mascheroni's geometry of the compass The Tetracyclic Plane: Fundamental theorems and definitions Cyclics The Sphere in Elementary Geometry: Miscellaneous elementary theorems Coaxal systems The Sphere in Cartesian Geometry: Coordinate systems Identity of Darboux and Frobenius Analytic systems of spheres Pentaspherical Space: Fundamental definitions and theorems Cyclides Circle Transformations: General theory Analytic treatment Continuous groups of transformations Sphere Transformations: General theory Continuous groups The Oriented Circle: Elementary geometrical theory Analytic treatment Laguerre transformations Continuous groups Hypercyclics The oriented circle treated directly The Oriented Sphere: Elementary geometrical theorems Analytic treatment The hypercyclide The oriented sphere treated directly Line-sphere transformation Complexes of oriented spheres Circles Orthogonal to One Sphere: Relations of two circles Circles orthogonal to one sphere Systems of circle crosses Circles in Space, Algebraic Systems: Coordinates and identities Linear systems of circles Other simple systems The Oriented Circle in Space: Fundamental relations Linear systems The Laguerre method of representing imaginary points Differential Geometry of Circle Systems: Differential geometry of thecircles Parametric method for circle congruences The Kummer method Complexes of circles Subject index Index of proper names.

Abstract: Recently, Tarnai and Gaspar [22] used mechanically inspired computer simulations to construct thin coverings of a square with up to ten equal circles. We generalise the problem to rectangles and determine the thinnest coverings of a general rectangle with up to five equal circles. Partial results are presented for coverings with seven circles.