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!
I will just assume that you forgot to put finite in your first sentence.
Then, your approach is fine (after adding a short comment why translations, and rotations with different centers are excluded).
It's hard to see from your question why the final step "took quite a long time".
You take the rotation with the minimal angle and any other one. Then, either the other one is already a multiple of the minimal one, or do a single step in the Euclidean algorithm (that is, subtracting the smaller angle from the larger one as often as possible) to get a contradiction to the minimality.
So, think that your difficulty is purely technical in the sense that you didn't write down your own argument in the simplest way. I don't even think that this last step would need more space the properly writing out the short comment why you can just regard rotations around a single point.