New singular points of triangles in connection with gnomonic projections of crystallographic complexes
Abstract
A large number of singular points are known, the positions of which are strictly derived for each given triangle. These points are discovered through the study of various properties of triangles, the number of which is very considerable. Despite all the simplicity of such a figure as an ordinary triangle, its study cannot yet be considered exhausted, and over time, although rarely, new properties are discovered. The problem arises to find the point that serves as the center of the linear prima of rays — the polars of points of the circumscribed circle. To solve this problem, it suffices to find the polars of any two points of the circle with respect to the triangle; the point of their intersection is the desired one. It is precisely this that constitutes the new singular point of the triangle referred to in the title.
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