|Dept.||Sobolev Institute of Mathematics, Novosibirsk State University|
The starting point of the lecture is the classical Heron’s formula that relates the area of a triangle to its side lengths.We discuss its non-euclidean versions know for a long time. Then we describe possible statements of the Brahmagupta’s formula for area of an inscribed quadrilateral on the Euclidean, spherical and hyperbolic planes. The natural generalization of these results is the Bretschneider for area of an arbitrary quadrilateral. The next subject will be the Pythagorus theorem for triangles and tetrahedrons in all the three spaces of constant curvature. We discuss also some other identities for triangles and tetrahedrons with addition properties.