On the Miquel point of simplices
Let \(S:=S({\mathbf x}_0,{\mathbf x}_1,\dots,{\mathbf x}_d)\) be a \(d\)-simplex with positive volume of the real Euclidean \(d\)-space, \(d\geq 2\); if one point is marked on each of the \(d(d+1)/2\) edges of \(S\) and a sphere \(M_i\) is drawn through each vertex \({\mathbf x}_i\) and the points marked on the \(d\) edges which meet in \({\mathbf x}_i\), then these \(d+1\) Miquel spheres \(M_i\), \(i=0,1,\dots,d\), all meet in a point \(M\) (Miquel's point). For this multidimensional generalization of Miquel's theorem (\(d=2\)) the author gives a rigorous analytical proof. As a by-product the author obtains a family of upper bounds of Gram's determinant. Two figures accompany the elegant and careful considerations.
scientific article; zbMATH DE number 5322349