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.

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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.