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Singular matrices and pairwise-tangent circles

Published online by Cambridge University Press:  15 February 2024

A. F. Beardon*
Centre for Mathematical Sciences University of Cambridge, Wilberforce Road, Cambridge CB3 0WB e-mail:


The idea of using the generalised inverse of a singular matrix A to solve the matrix equation Ax = 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 (v1, …, vn) of its distinct, consecutive, vertices. It is convenient to let vn + 1 = v1 and (later) Cn + 1 = C1.

© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

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