Let $C$ be a Jordan curve in $\mathbb{R}^2$. Does there exist points $P,Q,R,S$ on $C$ such that quadrangle $PQRS$ is a non-degenerate rectangle?
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7$\begingroup$ Yes, see Matschke's survey. Search the document for "Inscribed rectangles". $\endgroup$– eins6180Aug 10, 2016 at 10:11
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$\begingroup$ Thanks. I didn't know that this problem has a name. $\endgroup$– user80247Aug 10, 2016 at 10:46
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$\begingroup$ The link in eins6180 comment seems to be dead. I will add Internet Archive link and also link to A Survey on the Square Peg Problem on Notices website. Wikipedia article on this: Inscribed square problem. $\endgroup$– Martin SleziakJan 16, 2018 at 5:23
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