Does this prove the Inscribed Rectangle/Square Theorem?

Question

1. Imagine a circle, with four points on it forming a rectangle.
2. Twisting and bending this circle along those four points, any curve can be produced. $\blacksquare$

Is this a proof of the Inscribed Rectangle Theorem? I thought this up the other day and it seems fine to me.

If this is actually a proof, then isn't it possible that the four original points could be forming a square? Wouldn't it prove the Inscribed Square problem too?