Stromquist's Theorem: If the simple closed curve J is "nice enough" then it has an inscribed square.
"Nice enough" includes polygons.
Read more about it here: www.webpages.uidaho.edu/~markn/squares
An "inscribed square" means that the corners of a square overlap with the curve.
I would like to suggest a counter-example:
The curve connected by the points:
Link to plot: http://www.freeimagehosting.net/uploads/5b289e6824.png
Can this curve be incribed by a square?
(An older version of this question had another example: a triangle on top of a square (without their mutual side.) )