ПРИМЕНЕНИЕ ПРИНЦИПА ДВОЙСТВЕННОСТИ ПРИ СОСТАВЛЕНИИ УПРАЖНЕНИЙ НА ЗАНЯТИЯХ ПО ГЕОМЕТРИИ

Abstract

Geometry is one of the complex disciplines where many facts are interconnected. It is possible to develop the idea of facts interrelations through correlation using the duality principle. The duality principle is known in projective geometry, mathematical logic. This principle is clearly pronounced in one of the theorems of new triangle geometry. The traditional analytical geometry course does not study the facts of new triangle geometry. To reinforce many topics of the analytical geometry course, for example, “The distance between two points”, “The symmetrical form of the equation of a line”, “The angle between two lines”, it is reasonable to consider some facts from the new triangle geometry in the Cartesian coordinate system. Thus, an element of novelty is introduced to the reviewed material. The guidebooks on triangle geometry solve tasks through classical approaches or applying barycentric coordinates not using analytical geometry formulas. The paper proposes the constructing technique for the couples of exercises using the duality principle in the plane geometry teaching methods. Tasks are constructed for the Cartesian coordinate system as this allows demonstrating the duality of points in the drawings. In the composed exercises, two drawings are constructed in parallel columns. In different cases, the points can be the triangle-apexes, an orthocenter, or a height base. The initial triangle sides are located on the axes of coordinates, and their side lengths set up Pythagorean triple for better understanding the task-solving algorithm by the students. The symmetrical form of the equation of a line shows the necessity of analytical study since it is difficult to check the distance from the orthocenter to the orthotriangle sides in the drawings due to the small value. For many such information units, the aggregation relationships (whole-part) are set up, reflecting the geometric embedding of components.