Duality Principle in Geometry

It is known that geometry originated and developed in connection with people's practical activities, measurements and construction. The material of the school geometry course is based exclusively on Euclidean geometry. However, students may have a legitimate question: after all, the surface of the Earth on which we live is not smooth, as assumed in Euclidean geometry. We must also take into account that the Earth is spherical in shape and that this sphere itself has protrusions, mountains, valleys and depressions on its surface. Or the area of land we want to measure is not a smooth surface; there may be certain depressions and curvatures on this surface.The research work is devoted to studying ways to improve the secondary school geometry course and the use of the principle of duality. Teaching geometry is fraught with a number of difficulties arising from the very structure of the course, or more precisely, from the existence of axiom systems and the correct determination of the degree of use of both logic and intuition in geometry. These problems can only be solved by informing students about the existence of non-Euclidean geometries.