Abstract:

The article continues the series of contributions focused on Apollonius' problems and their possible solutions at various levels of education. The focus on this type of problems stems from their interest and suitability for inclusion in mathematics education and in prospective mathematics teachers' training, because just as new mathematical knowledge was discovered when solving these problems in the past, so now students can also research, inquiry and discover as part of their undergraduate training. Contemporary education includes a number of new trends that can be successfully applied in solving these problems. Whether it is a constructivist approach, e.g. inquiry-based teaching or the use of computers and appropriate software in education, and thus at the same time supporting the development of students' digital literacy.

Unfortunately, Apollonius' problems are not among the standard problems in mathematics education. The reason is probably that some of these problems require mathematical knowledge to solve, which is not taught in secondary education. E.g. the selected Apollonius problem LCC (line-circle-circle) is usually solved using circular inversion, which is only introduced to students in tertiary education. This is a shame, because circular inversion is not such difficult mathematics, especially nowadays with the possibilities of using ICT in teaching.

As part of the verification, the application of Apollonius' problems in teaching mathematics was initially a problem for students (prospective mathematics teachers) as well. In addition to circular inversion, the students also did not know dilatation, another mathematical tool used to solve an LCC-type problem. The advantage for students is always that they already know how to use ICT, and especially the Geogebra software to support geometry, so they can get to know new geometric knowledge faster and even build and verify some of them themselves thanks to the interactivity and dynamism of this geometric software.

A frequent tool for solving geometric problems is the use of FPS (Fixed Point Sets) method. Students know the sets of points such as conic sections, but they can't use them in solving geometric problems, because they can't construct them manually. The new era brings new possibilities and new methods of solving familiar problems can also arise. Of course, these new methods come mainly from the use of ICT and above all suitable software. After the Covid-19 epidemic, the increased use of ICT in education still has inertia, and hopefully it will remain so, because a certain transformation of education must occur through the development of society and technology even without epidemics.

The students were also shown the possibility of trying to solve Apollonius' problem LCC using the FPS method, specifically with the help of conic sections. The analysis of the specified problem revealed the need to use hyperbolas and parabolas. In the article, we present the solution of Apollonius' problem LCC with the help of these conic sections.

If students (prospective mathematics teachers) are able to solve problems standardly solved by university knowledge thanks to ICT already with the mathematical knowledge of the secondary school, they will not only be able to show their pupils interesting mathematical problems earlier, but they themselves and their pupils will be aware of the importance and benefit of digital technologies for support for education and their own development.

Keywords:

Apollonius’ problems, Apollonius’ problem LLC, Circular Inversion, Dilatation Conic Sections, ICT, Geogebra.