In https://arxiv.org/abs/2002.05422 I proved with elementary topological method that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a non-zero number of times) can always be split into 3 arcs rearrangeable to a closed smooth curve. Here comes one example of what I called the 2-cut theorem.
I am now writing the introduction to my thesis and I would like to cite more examples of counterintuitive properties of planar closed curves, like the one above I think it is. It came already to my mind the inscribed square problem, asking whether every Jordan curve admits an inscribed square (pic from https://mathoverflow.net/questions/275439/jordan-curves-admitting-only-acyclic-inscriptions-of-squares).
Such a property is still a conjecture for the general case, but proofs have been provided for several special cases and the easier inscribed rectangle problem can be solved with a beautiful topological argument (3Blue1Brown made a very nice video about that https://www.youtube.com/watch?v=AmgkSdhK4K8&t=169s).
My question: what are other surprising properties of closed planar curves you are aware of?