Abstract:

The article continues in a series of contributions focused on Apollonius' problems and their possible solutions at various levels of education. As a part of the undergraduate prospective teachers training of mathematics for secondary education (ISCED 2, ISCED 3), we also deal with these problems, as some of them can be solved on the basis of knowledge of school mathematics in secondary education and, unfortunately, they are not among the standard problems in teaching mathematics. We use "unfortunately" mainly because these problems are very interesting, especially because their solution requires various mathematical knowledge and relationships. Pupils can apply solutions to these problems. Pupils can apply in solving these problems Fixed Point Sets, Homothety, Power of a Point Theorem, Radical axis, Radical center and more. Students meet Apollonius' problems for the first time in college (ISCED 6) probably because Circular inversion is used as a universal tool for solving Apollonius' problems, with which students will get acquainted only in tertiary education. Historical notes about Apollonius of Perga and his best-known works Konika and De Tactionibus are also associated with these problems, thus promoting interdisciplinary relationships in teaching. Following the work Konika, the Conic sections will be mentioned again.

Apollonius' problems are solved with students by the above standard methods, but these problems also allow some experimentation. Various factors interfere in the educational process: modern trends in education (constructivist approaches, inquiry-based teaching, use of ICT, development of digital literacy, etc.), but also factors such as the Covid-19 epidemic in which education had to be switched to distance online teaching. This further strengthened the role of ICT in education, which supported new methods of solving problems using ICT. There has been more use of mathematical software in teaching mathematics, especially those tools that are available online and don’t need to be installed on a PC. From mathematical software we can recommend Wolfram Cloud for the calculations and Geogebra for geometric constructions. Apollonius' problems are mainly positional problems within geometry. By teaching online using the tools of dynamic geometry Geogebra, it was possible to try out an inquiry-based approach to teaching and let students look for "new" non-standard solutions to these problems. As a "new" non-standard solution method, students, under a certain teacher's guidance, eventually saw the possibility of using conic sections. This method is certainly not one of the standard methods for solving Apollonius' problems, because on paper this method is impracticable. These solution methods are only practicable using the suitable software, which Geogebra is.

In this paper, we will focus on two Apollonius' problems such as problem PLC (point, line, circle) and problem PCC (point, circle, circle), which are standardly solvable with the help of Circular inversion, but students were able to solve them in other ways. As already mentioned, students encounter Circular inversion only within tertiary education, therefore, it is not possible to solve these problems with pupils earlier. However, if they are able to solve them with the help of Conic sections, they will be able to solve these problems in their pedagogical practice already in secondary education. Thus, ICT not only makes it possible to solve mathematical problems more easily, faster, but also makes it possible to bring new knowledge to pupils earlier than it would be without the use of ICT.

Keywords:

Apollonius’ problems, Apollonius’ problem PLC (point-line-circle), Apollonius’ problem PCC, Circular Inversion, Conic Sections, ICT, online teaching, Geogebra.