Equilateral Triangles Inscribed in a Parabola

Author:
Roy Wright
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.
On paper, without technological help, see if you can find the exact coordinates for an equilateral triangle whose vertices all lie along the graph of . Could you do this if a different vertex along the graph was chosen for you to begin with?