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Equilateral Triangles Inscribed in a Parabola

A question from Twitter: Is it possible to choose points on the graph of to form vertices of an equilateral triangle? The red point shown below can be dragged along the parabola . It is always possible to find two other points on the parabola to form an equilateral triangle. In fact, in many cases, it is possible to do this three times! The checkbox labeled "show more" gives a sort of visual hint as to how the triangle(s) can be found geometrically.