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Geometry & Graphics

AN INFINITE SEQUENCE OF MUTUALLY RELATED CIRCLES An Infinite Sequence of Mutually Related Circles

Published in Geometry & Graphics · Volume 4, Issue 3 · Pages 29–35 · Rubric: Scientific problems of geometry
DOI: https://doi.org/10.12737/21531
Received: 19.09.2016 Accepted: 19.09.2016 Published: 19.09.2016 Language of publication: RUS
Authors
A.
A. Arakelyan
ABSTRACT
Geometry problems on the circle was discussed by geometers for thousands of years. The first task with different configurations concerning circles mutually meet in the works of Archimedes, Apollonius Pergskiy and Puppa Alexandria. Later Rene Descartes in his letters to Princess Elisabeth of Bohemia discusses some of the challenges for dealing with circumference. In particular, named after him Descartes Theorem asserts that the radii of four circles mutually related, any three of which do not have a common tangent, square satisfy some equation. This equation and some of its consequences were known to mathematicians of ancient Greece more than two thousand years ago (for example, the problem of Apollonius Pergskiy about how to build circles for the three given circles). Solving this equation, we can construct a fourth circle tangent to three other circles of the set. The article deals with infinite sequence of mutually dealing with circles inscribed in different configurations of Descartes. For each case obtained the interesting algebraic relations for relations of the constituent circles. As a practical application of the results of solved classical problem for Puppa arbelos and considered three different infinite sequence of circles inscribed in the arbelos. For radius´s of the circles of these sequences the relations expressed through the radii of circles of the arbelos. Since the arbelos figure, in a sense, can be seen as the contours of cyclide Dupin, the results can also be useful in the study of the properties inscribed spheres in the channels a special cichlid Dupin.
KEYWORDS
descriptive geometry, engineering geometry, modeling, synthetic and analytical methods, multidimensional forms.
Text References
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Введение

Окружность как совершенная фигура всегда привлекала внимание геометров своей простотой, изяществом и бесконечным таинством. Несмотря на то что исследование окружности само по себе весьма интересна, тем не менее возникают множество замечательных возможностей при расмотрении конфигураций окружностей относительно друг друга. Эти, казалось бы, простые, но на самом деле довольно сложные задачи классической геометрии, часто содержат неожиданные факты. Геометрические задачи на касающиеся окружности обсуждались в течение тысячелетий. Первые результаты, связанные с различными конфигурациями взаимно касающихся окружностей, являются работы Архимеда, Апполония, Паппа и Декарта [2–4, 11, 16, 17]. В дальнейшем также выявлено множество знаменательных свойств для последовательностей взаимно касающихся окружностей. В частности, стоит отметить работы Штайнера, Форда, Кокстера и др., которые предлагали методы как классической, так и не классической геометрии [1, 9, 15, 19–21, 24, 25].

В данной статье рассмaтриваются разные последовательности взаимно касающихся окружностей. Для каждого случая получены интересные геометрические и алгебраические соотношения для взаимосвязей составляющих окружностей, которые дают возможность почувствовать красоту геометрии.

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