Singular matrices and pairwise-tangent circles

The idea of using the generalised inverse of a singular matrix A to solve the matrix equation A x = b has been discussed in the earlier papers [1, 2, 3, 4] in the Gazette . Here we discuss three simple geometric questions which are of interest in their own right, and which illustrate the use of the generalised inverse of a matrix. The three questions are about polygons and circles in the Euclidean plane. We need not assume that a polygon is a simple closed curve, nor that it is convex: indeed, abstractly, a polygon is just a finite sequence ( v 1 , …, v n ) of its distinct, consecutive, vertices. It is convenient to let v n + 1 = v 1 and (later) C n + 1 = C 1 .