ARBELOS AND ASSOCIATED CIRCLES

Abstract

Geometry as a science originated in Ancient Greece, its axiomatic constructions are described in the "Elements" of Euclid. Euclidean geometry studied the simplest figures on the plane and in space. Greek-speaking mathematicians who lived between the 6th century BC and 5th century AD, posed and solved many interesting geometric problems. Most of these tasks were solved graphically, which required the execution of a large number of various complex constructions. At that time it was believed that "truly geometric" are those tasks that were solved only with the help of such "scientific instruments" as a compass and a ruler. Ancient Greek mathematicians paid special attention to one of the most important geometric figures – circles, which even in those days were widely used in practice. Archimedes of Syracuse, who first introduced the concept of arbelos, made a significant contribution to the study of the circle. By arbelos, he understood a flat geometric figure formed by a certain semicircle, from which two smaller semicircles with diameters lying on the diameter of the original semicircle were cut out and divided into two parts. Thus, a curvilinear triangle was formed, bounded by three semicircles. This work considers the issue of solving well-known ancient geometric problems using modern methods of engineering graphics, analytical geometry and numerical methods, without additional constructions, which are used in the graphical solution of the problems under consideration. When implemented numerically, the problem was reduced to solving a nonlinear equation with one unknown. Nonlinear equations are associated with finding the radii of inscribed or circumscribed circles and the coordinates of their centers. In the work, in particular, circles inscribed in arbelos, paired circles of Archimedes, known as twin circles, the Pappa chain of Alexandria are constructed. Based on the research of modern mathematicians on arbelos, the problems of constructing the circles of Bankoff, Schoch, Woo were solved.