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A Jordan curve is a plane curve which is topologically equivalent to (a Homeomorphic image of) the Unit Circle.
It is not known if every Jordan curve contains all four Vertices of some Square,
but it has been proven true for ``sufficiently smooth'' curves and closed convex curves (Schnirelmann). For every
Triangle 
and Jordan curve 
, has an Inscribed Triangle similar to .
See also Jordan Curve Theorem, Unit Circle