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Here's a nice 3blue1brown video about the easier version of the inscribed rectangle problem, that some inscribed rectangle exists: https://www.youtube.com/watch?v=AmgkSdhK4K8

And I remember thinking this was a nice overview of some aspects of the "square peg problem" (the Toeplitz conjecture): https://www.ams.org/notices/201404/rnoti-p346.pdf

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