Application of the Noncommutative Gröbner Bases Method for Proving Geometrical Statements in Coordinate-free Form

In this paper we consider the application of the noncommutative Gröbner bases method for proving theorems in algebraic geometry. Geometrical statements of constructive type should be given in the coordinate-free form. 1. Coordinate-free representation of points and statements We consider theorems of elementary geometry (two-dimensional and three-dimensional). Let A1, A2, A3, . . . , An be points in a finite-dimensional space. We treat these points as vectors drawn from the origin 0. Then, geometrically, the outer product of two vectors A and B is a bivector corresponding to the parallelogram obtained by sweeping the vector A along the vector B. The parallelogram obtained by sweeping B along A differs from the parallelogram obtained by sweeping A along B only in the orientation. Consider the Grassman algebra generated by points A1, A2, A3, . . . , An, i.e., the free algebra with an external product A ∧ B, which is associative and anticommutative: A ∧B = −B ∧ A. Consider a finite-dimensional space and task-space embedded in this space. For example, in the case of a two-dimensional task we consider a plane in the enveloping space.