Apollonius Solutions in Rd

A TAXICAB VERSION OF A TRIANGLE' S APOLLONIUS CIRCLE

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...

On Three Circles

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...

Apollonius tenth problem via radius adjustment and Möbius transformations

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...

Spellbinding Performance: Poet as Witch in Theocritus' Second Idyll and Apollonius' Argonautica

The Farey structure of the Gaussian integers

Arrange three circles so that every pair is mutually tangent. Is it possible to add another, tangent to all three? The answer, as described by Apollonius of Perga in Hellenistic Greece, is yes, and, indeed, there are exactly two solutions [oP71, Problem XIV, p.12]. The four resulting circles are called a Descartes quadruple, and it is impossible to add a fifth. There is a remarkable relationshi...