The theorem of symmetric rotations and its corollary can be proved
easily.(Proof left to the reader)
But no rigorous proof(?) for the
iterative trisection construction based
on
them.
_____________________________________________________________________
An
interesting definition of the circle :
A circle is the locus of a
moving point with respect to two fixed points,in the same plane,
such that
the square of the distance between the fixed points is the sum
of the squares
of the distances between the moving point and the two
fixed
points.