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A rotation about the origin, through angle phi, is easily expressed in
terms of polar coordiantes by (r, theta) gets mapped to (r, theta +
phi). Use some trigonometric identities to show that, for an
appropriate choice of phi, this is the same as the cartesian mapping (x,y)
to (-x,-y).
If the isometry has more than one invariant point, it must be a
reflection - why? If A is mapped to B, where is B mapped to? What can
you say about the image of the mid point of A and B? Use these
suggestions to show that there is at least one invariant point. Then
show that it must be a half turn if there is exactly one invariant
point.
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