Perspective simplexes and tucker quadrics

Abstract

The sides of a triangle are met by the sides of a triangle in perspective with it in nine points of which three lie on the axis of perspectivity of the triangles and the other six, as we have it from the converse of Pascal's theorem, lie on a conic. When the centre of perspectivity is the Lemoine point of the triangle and the axis of perspectivity is the line at infinity this conic is a circle known as a Tucker circle [1]. In this paper, analogously we prove that given a simplex inn-space, the traces on its edges of the prime faces of any simplex in perspective with it, all except those falling on the prime of perspectivity, lie on a quadric which we call a Tucker quadric. And further we consider the special case when this quadric becomes a hypersphere.