Polar relations of imaginary triangles and tetrahedrons

Abstract

The known properties of the gnomonic projections of trigonaloid crystals prompted me about the presence of the relationships mentioned in the title that seemed paradoxical to me. For the case under consideration, the theory of poles and polars unfolds in its usual form: two points are the poles of two polars and, in turn, determine the straight line-polar of the point of intersection of these polars. Each vertex of a trigon has a polar opposite side, etc., and in no case there is a point through which its polar would pass, as it occurs in the case for imaginary projective conjectures (see article).