mg.metric geometry - Can the circle be characterized by the following property?

If a figure had an axis of symmetry in three non-parallel
but non-concurrent axes, then composing these suitably would
give a translative symmetry, which is impossible if the figure
is bounded. So all the axes of symmetry of your putative curve
are concurrent through a point $O$ which we shall call a centre.
Then all rotations about the centre $O$ are symmetries. The only
simple closed curves with this property are circles centred at $O$.