One of the most famous problems of classical geometry is the Apollonius' problem asks construction of a circle which is tangent to three given objects. These objects are usually taken as points, lines, and circles. This well known problem was posed by Apollonius of Perga ( about 262 - 190 B.C.) who was a Greek mathematician known as the great geometer of ancient times after Euclid and Archimede...

We study the external tritangent Apollonius (or Voronoi) circle to three ellipses. This problem arises when one wishes to compute the Apollonius (or Voronoi) diagram of a set of ellipses, but is also of independent interest in enumerative geometry. This paper is restricted to non-intersecting ellipses, but the extension to arbitrary ellipses is possible. We propose an efficient representation o...

We provide a new solution for the famous Apollonius Tenth Problem. The problem is to construct a circle tangent to three given circles in the plane. An equivalence between analytical and geometric approaches to the solution is explored. A new analytical solution for the problem is presented and a step by step geometrical interpretation of this solution is provided. We show that this interpretat...

The Apollonius Tenth Problem, as defined by Apollonius of Perga circa 200 B.C., has been useful for various applications in addition to its theoretical interest. Even though particular cases have been handled previously, a general framework for the problem has never been reported. Presented in this paper is a theory to handle the Apollonius Tenth Problem by characterizing the spatial relationsh...

Abstract. The classical Three-Circle Problem of Apollonius requires the construction of a fourth circle tangent to three given circles in the Euclidean plane. For circles in general position this may admit as many as eight solutions or even no solutions at all. Clearly, an “experimental” approach is unlikely to solve the problem, but, surprisingly, it leads to a more general theorem. Here we co...

The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i], inspired by, and intimately related to, the ‘sensual qu...

In this paper, we consider a multi-pursuer single-superior-evader pursuit-evasion game where the evader has a speed that is similar to or higher than the speed of each pursuer. A new fuzzy reinforcement learning algorithm is proposed in this work. The proposed algorithm uses the well-known Apollonius circle mechanism to define the capture region of the learning pursuer based on its location and...

The variance of a weighted collection of points is used to prove classical theorems of geometry concerning homogeneous quadratic functions of length (Apollonius, Feuerbach, Ptolemy, Stewart) and to deduce some of the theory of major triangle centers. We also show how a formula for the distance of the incenter to the reflection of the centroid in the nine-point center enables one to simplify Eul...

The Apollonius Circle Problem dates to Greek antiquity, circa 250 BC. Given three circles in the plane, find or construct a circle tangent to all three. This was generalized by replacing some circles with straight lines. Vieta [19] (or Viète) solved the problem using circle inversions before 1580. Two generations later, Descartes considered a special case in which all four circles are mutually ...

The Apollonius Circle Problem dates to Greek antiquity, circa 250 b.c. Given three circles in the plane, find or construct a circle tangent to all three. This was generalized by replacing some circles with straight lines. Viéte [Canon mathematicus seu Ad triangula cum adpendicibus, Lutetiae: Apud Ioannem Mettayer, Mathematicis typographum regium, sub signo D. Ioannis, regione Collegij Laodicens...