Abstract:

The article continues in a series of contributions focused on Apollonius' problems and their possible solutions in secondary education (ISCED 2 and ISCED 3) with a demonstration of new possibilities thanks to the use of ICT. The first contribution “Apollonius’ problems in secondary education using ICT“ in EDULEARN20 Proceedings 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, three of which (PPP, LLL and PLL) were resolved in the paper. 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 a specific country. This provided the prerequisites for solving these problems. Prerequisites for new methods of solution were given by the use of ICT.

The second paper “Apollonius’ problem LLC at high school using ICT“ in ICERI2020 Proceedings focused on the fourth Apollonius' problem, which can be solved in secondary education, specifically to the problem LLC that pupils are able to solve with the standard method using homothety but also thanks to ICT and GeoGebra Software, we have shown a non-standard solution using parabolas. Again, it was pointed out the compliance with the mathematics curriculum that pupils have the necessary apparatus (knowledge of homothety and Fixed Point Sets - conic sections) to solve this problem.

In this article, we will focus on the last two Apollonius’ problems, which can still be standardly solved with high school knowledge of mathematics. It is a type PPL and PPC of Apollonius' problems. These problems can be standardly solved using knowledge of the power of a point theorem and radical center. We will also point out a non-standard solution of these problems by the method using Fixed Point Sets - conic sections, which is again possible only with the help of ICT and appropriate software (eg. GeoGebra).

With this article, we will again point out the current trends in mathematics education, such as assigning non-standard problems, interdisciplinary relationships and above all, the benefits of using ICT and supporting the development of pupils' digital literacy.

Keywords:

Apollonius’ problems, Apollonius’ problem PPL and PPC, secondary education, ICT, conic sections.