COORDINATE METHOD IN THE PROBLEMS OF HIGHER COMPLEXITY OF GEOMETRIC CONTENT

Abstract

The state of methodological support for the preparation of students of higher educational institutions of non-core specialties for participation in the olympiad in the discipline "mathematics" is examined in the article. The Olympiad problems of geometric content are most difficult to solve traditionally. Difficulties arise due to the need to perform additional constructions and establish complex relationships between elements of geometric figures and bodies. In addition, geometric problems of the olympiad level, as a rule, require methods of several branches of mathematics for their solution.

In this case the coordinate method reduces the cognitive complexity of the solving process. Such a process is easier to algorithmize. It brings the coordinate method to algebraic methods.

The efficiency of solving geometric problems by the coordinate method substantially depends on the appropriate placement of the studied figure or body in the coordinate system. In problems where it comes to figures inscribed in a circle, it is advisable to use the relationship between the Cartesian and polar coordinate systems. For the calculating of figure area, one can use formulas containing determinants with the coordinates of the vertices of the triangles that are parts of them. This technique, combined with the coordinates of the vertices, which are expressed in terms of the polar radius and polar angle, allows the use of trigonometric identities to simplify the resulting expressions.

Additional opportunities for the development of students' academic search abilities are provided by conditional optimization problems, where cases of coincidence and difference between global and conditional extremes are possible.

The article considers the geometric problem that was proposed at the international Olympiad for students. For this problem, the author's solution by the coordinate method is presented. In published solutions for this problem, this approach was not used. The studied problem can be formulated in terms of the problem of conditional optimization. A feature of its solution is the fact that one of the points of global maxima satisfies the existing restrictions.

Key words: coordinate method, olympiad in the discipline "mathematics".