A Jordan curve is a plane curve which is topologically equivalent to (a homeomorphic image of) the unit circle, i.e., it is simple and closed. It is not known if every Jordan curve contains all four polygon vertices of some square, but it has been proven true for "sufficiently smooth" curves and closed convex curves. For every triangle T and Jordan curve J, J has an inscribed triangle similar to T.

Carathéodory's theorem | closed curve | Jordan curve theorem | simple curve | square inscribing | unit circle