I recently stumbled upon an interesting property of ellipses that I had not seen before. I can prove it with not much trouble by analytic geometry. But unfortunately my proof does not offer much in terms of insight. Can anyone come up with an "insightful" demonstration of why this is true?
Proposition: The length of the diagonal of a circumscribing rectangle of an ellipse is independent of the rectangle's orientation.
Corollary: The locus of vertices of all circumscribing rectangles of an ellipse is a circle.
Note 1: A "circumscribing rectangle of an ellipse" is a rectangle with all its four sides tangent to the ellipse. (The sides of the rectangle need not be parallel with the axes of the ellipse.)
Note 2: If the major and minor semiaxes of the ellipse have lengths a and b, it can be shown that the length of the diagonal of the circumscribing rectangle is twice sqrt(a^2+b^2).