Obtaining a 3D extension of Pascal theorem for non-degenerated quadrics and its complete configuration with the aid of a computer algebra system

Abstract

Pascal's classic theorem states that: "the three intersection points of opposite sides of a closed hexagonal line, inscribed in a nondegerated conic, are collinear". The following extension of Pascal theorem to 3D is considered: "given a closed decagonal line, inscribed in a nondegerated quadric, whose opposite side-lines are secant, the five intersection points of opposite side-lines are coplanary". (A polygonal line with 10 sides is considered, because 10 - 1 points determine a quadric, as 6 - 1 points determine a conic). Obviously, in this extension to 3D of Pascal theorem some vertices of the polygonal line can not be freely chosen, but an interesting property has been found: the five diagonal lines passing through opposite vertices share a point. This property leads to a simple method to generate the configuration. Moreover, conditions of existence of this configuration are determined and the so called complete configuration is also described in detail. As large expressions appear when coordinates are used, we have developed a package on a computer algebra system that helps us to find and to automatically generate this configuration.