Invariant points in the product of four inversions of non-coplanar centers
By using the Clifford algebra of \(R^n (n\geq 3)\), a real \(n\)-dimensional vector space equiped with Euclidean metric the author proves that if \(S_i\), \(i=1, \dots,4\) are centers of inversion spheres, then the straight line \(S_1S_4\) is the common perpendicular to the straight lines \(S_1S_2\) and \(S_3S_4\) which are orthogonal. He shows that there exist invariant points for \(n>3\) in the subspace orthogonal to the centers' subspace if the power \(k\) of four inversions is upper to \(3a^2/4\) and if the lengths of \(S_1S_2, S_1S_4, S_3 S_4\) are equal to \(a\). When \(k= 3a^2/4\) there exists in the center's subspace an invariant point which is equidistant of four centers', being common to the four inversion's spheres. When \(k<3 a^2/4\) no invariant point exists.
scientific article; zbMATH DE number 1156793