See this question: mathoverflow.net/questions/306703/… it is somehow related to this problem and your answer.

@Gerry Myerson for equilateral triangle we have 3 two_thirds of a circle passing through its vertices, so it could not be an answer, just whole of a circle satisfies the condition not any part of it.

It seems that I have found something important about this problem, check the third answer

@erz Nice generalization you've made from my raw problem. Finally you've found an application here with these abstract curves and it may help to attract and makes more excitement. Also nice statement and arguments for special cases of the problem which lead us to better understanding about jordan curves and to see how amazing and complex they could be.so thumbs up. I'm concerning about the sauvage realm of non-convex cirves,hope to find some useful statements about them.