PYTHAGOREAN THEOREM IN VARIOUS GEOMETRIES

The work is devoted to the Pythagorean Theorem, known in school geometry, which expresses the relationship between the legs and the hypotenuse in a right triangle. This theorem is valid both in the elliptic plane (Riemann plane) and in the hyperbolic plane (Lobachevsky plane), in which triangles are considered as geodesics. It should be noted that the geodesic triangle in the Riemannian plane, the same as the spherical triangle, has sides that are arcs of a great circle. This is proven in differential geometry. To make it more clear, we emphasize that the orthogonal projections of such arcs in each tangent plane, drawn at any point of the arc, are straight line segments. A geodesic triangle in the Lobachevski plane (on the pseudo sphere) also has an pictorial commentary. It is noteworthy that the side of a geodesic triangle is an arc with minimal length. From differential geometry it is known that each of these planes has its own metric; a quadratic form I. In the work, using the known parametric equations of the sphere and methods of differential geometry, the Pythagorean Theorem, which was previously obtained from the corresponding Cosine theorem, is proved (formula (4)). Further, using Taylor series of trigonometric functions they show that when the Gaussian curvature of surfaces tends to zero, then the Euclidean case of the theorem is obtained. The scientific novelty of the work is the proof of this approach and there is also likely to be a connection with school geometry.