Abstract:

We consider Apollonius' problems at various levels of education to be very interesting and important for pupils to get acquainted with it, because these problems have always occupied a particularly important place among planimetric problems, because its solution requires knowledge of various mathematical relationships.

Another interesting feature of these problems is the fact that during its solution many new knowledge have been discovered over time and thanks to modern technologies (ICT), it is still possible to come up with new solution methods. Pupils get acquainted not only with Apollonius' problems and various mathematical knowledge, but when solving it with modern methods, they also develop their digital literacy.

The paper complements the article “APOLLONIUS’ PROBLEMS IN SECONDARY EDUCATION USING ICT” in EDULEARN20 Proceedings, which pointed out the four simplest Apollonius' problems solvable with knowledge of the geometry of secondary education. The pupils 2nd level of elementary school (lower secondary education, ISCED 2) they are able to solve Apollonius' problems PPP and LLL. Pupils of high school (upper secondary education, ISCED 3) they are able to solve Apollonius' problems PLL and LLC.

The previous paper focused on defining Apollonius' problems, historical notes on Apollonius of Perga and on his most famous works. The four simplest Apollonius' problems suitable for secondary education have been identified: PPP, LLL, PLL and LLC. In order to point out the solvability of these tasks in secondary education, it was necessary to show compliance with the mathematics curriculum according to the curricular documents of the Czech Republic. This provided the prerequisites for solving selected problems.

In lower secondary education, pupils are able to solve Apollonius' problems PPP and LLL using Fixed Point Sets (FPS), namely the use of circles (circumscribed triangle circle, inscribed triangle circle and escribed circles of a triangle) and lines (perpendicular bisector, angle bisector). In upper secondary education, pupils are able to solve Apollonius' problems PLL and LLC using homothety.

The use of ICT brings new possibilities for pupils to solve problems. To solve geometric problems are currently mainly used dynamic geometry tools – eg. GeoGebra. This tool has the potential to enable pupils already in secondary education to solve geometric problems more quickly, more precisely, more clearly and also allow them to use methods of solution that would be unrealistic without ICT. Therefore, we could show how to solve the Apollonius' problem PLL also by the FPS method, where the searched sets of points were parabolas.

Within the given range of the paper, it was not possible to show the solution of the last of the four selected Apollonius' problems suitable for secondary education, namely problem LLC. In this article, we will introduce a standard solution to the Apollonius' problem LLC realized by using homothety, but also thanks to ICT and SW GeoGebra we will also show the solution using the FPS method – parabolas. Until now, unless we can solve otherwise, circular inversion has been used as a universal tool for solving Apollonius' problems. If we accept for Fixed Point Sets not only differently defined lines and circles, which together with the point we classify as circular curves, also conic sections, we can say that every Apollonius' problem can be solved by FPS method.

Keywords:

Apollonius’ problems, Apollonius’ problem LLC, secondary education, ICT, conic sections.