PROOFS OF THE FUNDAMENTAL THEOREMS OF THE POINT CALCULUS

Abstract

The manuscript includes proofs of the fundamental theorems of the point calculus, on the basis of which the basic tools for modeling and parameterization of geometrical objects in the point calculus are developed. The basic theorem of simple relations or O-theorem is a generalization of Menelaus’ theorem for a polygon. Its proof in the point calculus is based on a particular case of the triangle area theorem, the essence and proof of which are also given. The triangle area theorem allows us to establish the relationship between the ratio of triangle areas and the simple ratios of three points obtained on the sides of the triangle. Another consequence of the theorem on the areas of triangles can be obtained as a result of using the theorem on the areas of triangles with equal altitudes, which confirms the reliability of the obtained results and their unambiguous relationship with the known theorems. The proof of the theorem on the ratio of the areas of triangles or S-theorem is based on the use of the properties of the determinants of the matrix and obtained by transforming them. This theorem has a generalization in the form of the theorem on the relation of tetrahedron volumes or the V-theorem of the point calculus with a further generalization to the multidimensional space, the proof of which is similar to that given in the manuscript. This makes it possible to determine areas, volumes and hyper-volumes of geometric objects using the denominator of the ratio as a unit of measure by means of simple relations of three points.