I feel it is the most natural and logical introduction to begin by mentioning that, in any field of activity, people are confronted with problems, some of them being solved easily, some requiring great efforts to search for the solution and others remain unsolved. At school, students encounter different problems, similar to those that will be encountered in their everyday lives, and thus they must be taught from early on how to distinguish the problematic situations, to put and formulate problems, to heuristically seek solutions and to solve them so that, ultimately, they can master the art of problem-solving.
In this day and age, the study of geometry is receiving significant attention, to a great extent due to its contribution to students’ cognitive abilities. Geometry also provides students with highly relevant transferable skills that are to last a lifetime, such as spatial orientation, critical thinking, problem-solving, quantitative and analytical skills, as well as the ability to construct logical arguments. Knowledge of some methods of reasoning in the study of geometry is also necessary because they facilitate a deep understanding of the multiple ways in which one can solve any problem, as well as the research this requires.
It is a well-known fact that geometry was born millennia ago, out of practical necessities. There is also often noted or distinguished a great stability of the truths and depths of geometry, which was true during the ancient Greek times and which is also true in the present. These features represent some essential notes of geometry that form an integral part of one’s general knowledge. However, the conclusion that geometry is a „complete science” is erroneous.
Moreover, collinearity and concurrence represent truths that are generally intuitive, but whose rigorous demonstration requires precise reasoning and a wide range of specific techniques. Thus, problems of collinearity and concurrence capture the students’ attention and develop their mathematical imagination, both in secondary school as well as in high school.
Undoubtedly, my recommendation is to combine the usual ordinary resolutions with those that focus on solving clustered issues or problems around a property with a central role in solving, problem-building on a given idea, using the heuristic methods and procedures in order to find the revealing idea; thus, being based on the use of active- participatory methods that place the student in the centre of the training process, with his questions and perplexity or bewilderment, with his real possibilities.