A Circle Covering Theorem
A. W. Goodman, Ruth Goodman · American Mathematical Monthly · 1945
THEOREM 1. Let the circles C1, , C,, with radii ri, * , r,, lie in a plane and have the following property: No line of the plane divides the circles into two non-empty sets without touching or intersecting at least one circle. Then the circles Ci, , C,, can be covered by a circle of radius r =n ir1. The theorem is trivial when n = 2. To the best of our. knowledge, however, no proof has been given for any n> 2. We shall give here an analytic proof of Theorem 1 which can easily be generalized to give the analogous theorem for euclidean space of k dimensions. This proof depends on the following lemma.