Longest parallel chord of an ellipse

Question

I am searching for a source demonstrating that, for any set of parallel chords spanning an ellipse, the longest chord passes through the center of the ellipse. I am not referring to the major and minor axes, which I know are the longest and shortest diameters. Rather, I am referring to any set of parallel chords and want to show that the longest chord is a diameter that passes through the center. This claim seems evident by visual inspection, but despite much searching, I cannot locate a source that establishes this claim analytically. I am writing an article in which this claim is relevant, so I would like to cite a source. Any sources would be much appreciated!

Answer 1

Every invertible linear transformation preserves the ratio of lengths of parallel line segments. Use a linear transformation that maps the ellipse to a circle.