How can we prove that any finite group of orientation-preserving isometries of the Euclidean plane is cyclic, using the following hint?
'You may assume that given any non-empty finite set E in the Euclidean plane, there is a unique smallest closed disc that contains E.'
I've found a proof not using this (the group must consist of rotations about a single point; pick the one with minimal angle of rotation and use Euclid's algorithm to show that this generates the group). But this took quite a long time, and I wondered if it would be quicker to use the hint.
Many thanks for any help with this!