A special simplex

Extract

Let S = A₀ hellip A_n be an n-simplex and A_ih the foot of its altitude from its vertex A_i to its opposite prime face S_i; O, G the circumcentre and centroid of S and O_i, G_i of S_i. Representing the position vector of a point P, referred to O, by p, Coxeter [2] defines the Monge pointM of S Collinear with O and G by the relation so that the Monge point M_i of S_i is given by If the n+1 vectors a are related by o_i be given by A_ih is given by Since A_ih lies in S_i, If T_i be a point on M_iA_ih such that i.e. That is, MT_i is parallel to oo_i or normal to S_i at T_i:. Or, the normals to the prime faces S_i of S at their points T_i concur at M. In fact, this property of M has been used to prove by induction [3] that an S-point S of S lies at M. Thus M = 5, M = S or .