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Oct 20, 2011 at 20:11 comment added Robert Bryant @Vladimir: Actually, that is exactly how I understood the question, and the answer is the same. The examples of convex curves $C$ (not all of them ellipses, by the way) that I indicate have the property that they have a circle of inscribed triangles of perimeter $L$, where $L$ the maximum value of the perimeter of any inscribed triangle in $C$. Moreover, for each point $P\in \partial C$, there is one inscribed triangle of perimeter $L$ passing through $P$. I don't see how this is different from what you are asking.
Oct 20, 2011 at 11:45 history answered Vladimir Georgiev CC BY-SA 3.0