Quadrics associated with a simplex in n-space

Abstract

It is well known that the projections of a pair of points from the vertices of a triangle onto the opposite sides lie on a conic and that when the points are the centroid and orthocentre of the triangle, this conic is a circle. Analogously the projections of the centroid and orthocentre of a simplex from its vertices onto the opposite (n—1)-dimensional faces, if the simplex is orthocentric, lie on a hypersphere [2, 5]. Further the projections of two points onto the edges of a general simplex from the opposite faces lie on quadric [1]; and when the points are the centroid and orthocentre respectively and the simplex is orthocentric, this quadric is a hypersphere [2]. The results as regards projections onto (n—l)-dimensional and 1-dimensional faces being thus known, it remains to see what results hold in the case of intermediary faces. And in this note we prove that a similar result holds for projections onto intermediary faces as well.