An application of the quadrilateral's geometry in solving competitive planimetric problems

Jordan Tabov, Asen Velchev, Rayna Alashka, Sevdalin Tsvetanov

In the present publication, which can be considered as a continuation of the paper V. Nenkov, St. Stefanov, H. Haimov, An application of quadrilateral's geometry in solving competitive mathematical problems, Synergetics and reflection in mathematics education, Proceedings of the anniversary international scientific conference, Pamporovo, October 16-18, pp. 121--128, 2020, the application of the geometry of quadrilateral to the solution of exams is considered. Three examples given in the magazine ``Mathematics and Informatics'' have been selected, the solutions of which illustrate well the benefit of studying the recently discovered properties of convex quadrilaterals. Two solutions to the tasks are presented for comparison. The first, proposed by participants in the competition, are relatively complex and longer, and the second---based precisely on elements of the geometry of quadrilateral, are significantly simpler and shorter. These solutions are based on properties of quadrilaterals associated with some of their remarkable points.