Is it true that parallelogram inscribed in ellipse is rectangle

Question

I only managed to prove that parallelogram inscribed in a circle is a rectangle.

Is it possible to prove that parallelogram inscribed in ellipse is rectangle? Maybe there are counterexamples?

I'm out of ideas.

Answer 1

Start with a rectangle inscribed into a circle. For simplicity imagine it to be aligned with coordinate axes. Apply an affine transformation to it, more specifically a shearing transformation. That will turn the right angles into something else, while at the same time preserving parallelism. It will also map conics to conics, and your original circle becomes an ellipse. So there you have it: a non-rectangular parallelogram inscribed into an ellipse.