Geomery problems with an implicitly defined structure – a means for developing intellectual skills in learners

The article is devoted to planimetric problems related to the midcenter of a triangle and the circle inscribed in it. The significant issue in these problems is, that more specific, additional reasoning is needed to determine their structure, because it is not explicitly define. On one problem, called „basic”, a detailed methodological development is made, having two goals. The first objective is to outline their „educational“ solutions comprehensively, intended to the purposeful preparation of students who will take the matriculation exam in mathematics or candidate student exam, that is why this problem has a criterion character. The other goal is to demonstrate a methodical technology for creating a series of new problems that derive from this basic problem. Different ideas for creating systems of problems are considered, and the corresponding formulations are presented.