[math-fun] Puzzle re maximum perimeter of triangle inscribed in ellipse

I recently heard an interesting question via the grapevine: Given an ellipse E in the plane, consider any point A of E. Then two other points B,C of E can be found so that the perimeter of the triangle ABC is maximized. Denote this maximum perimeter by P(A). Prove or disprove: P(A) is necessarily independent of A. --Dan