Abstract:

The article concludes a series of contributions focused on Apollonius' problems and their solution possibilities at different levels of education. Apollonius' problems have always occupied a particularly important place among planimetric problems. As part of the undergraduate training, we include these problems in the teaching, because when solving them, students can also inquiry and discover different methods for solving these problems. Students are also introduced to certain historical notes to know how old these problems are and surely have been solved many times. Nevertheless, they are introduced to the fact that within the framework of the development of digital technologies, the inclusion of ICT in education, including appropriate software, it is necessary in such a changing world to proceed with the transformation of the educational process, the use of new didactic aids and thus also discover new methods of solving even already solved problems.

For this purpose, we kept the so-called general Apollonius' problem at the end, which is considered the most complicated. The usual method of solving Apollonius' problem CCC (circle-circle-circle) is dilatation and circular inversion. The situation is also complicated by the number of possible configurations of input elements. According to the mutual position of the specified circles, there can be 49 configurations and based on these configurations, from 0 to infinite number of solutions. Students realize very quickly that solving these problems without ICT and suitable software is no longer meaningful today. Students were allowed to experiment and inquiry with the help of, for example, the Geogebra software. Students are familiar with transformations dilation and circular inversion from previous lessons. The greatest benefit of using Geogebra's interactive and dynamic geometry tools is that by changing the layout of input elements (free elements), all subsequent constructions (bound elements) are automatically redrawn. So, basically, one case or one problem configuration is representative of all possible configurations. Students could thus very quickly test all possible configurations of the chosen problem.

With further historical notes, the students were introduced to the most famous publications of Apollonius of Perga: Kónica and De Tactionibus. Since Apollonius was devoted to both conic sections and problems about tangents of circles, wouldn't it be possible to use these findings together and try to solve general Apollonius' problem CCC using conic sections? If so, ICT tools and appropriate software, which is Geogebra, will definitely be used for this purpose, because it is unrealistic to construct these curves accurately manually (on paper using a ruler and compass). As the first configuration, students were given 3 non-intersecting circles. From the analysis of the given problem, the need to use hyperbolas emerged. In the article, we will present the solution of the general Apollonius' problem CCC with the help of these conic sections – hyperbolas. For other configurations of this problem, other conic sections are also applied.

In this way, students (prospective mathematics teachers) can realize in their undergraduate training that with the development of societies and technologies, there is a need to transform education as well, implement new methods, new didactic aids, to develop one's competence and, subsequently, that of one's students, also by solving problems that have long been solved.

Keywords:

Apollonius’ problems, Apollonius’ problem CCC (circle-circle-circle), Circular Inversion, Dilatation Conic Sections, ICT, Geogebra.