Inscribing equilateral triangles in convex curves
Let C be a smooth, convex, closed curve, i.e., one without endpoints, and such that a line segment joining any two points on C lies inside C. (An ellipse or an oval are examples. "Smooth" means it has a tangent line at each point.)
Let P be any point on C. Show convincingly that you can always find two other points Q and R on C such that PQR is an equilateral triangle. (Try some sketches.)
Let us denote by D the region inside of C, and let TP denote the set of all equilateral triangles, one of the vertices of which being P. I considered then the set
I tried showing that S is a closed interval of the form [0,d], and then showing that a maximal triangle of side length d must be touching the curve at three points. However, I couldn't complete the argument.
I understand that being an introductory textbook, it is not expected from a reader to provide a rigorous proof here, but I'm curious to see what such a proof would like.