Dear Rolf,

Yes, this is indeed the answer that I was
hoping for:

  The midpoints M1 and M2 of parallel chords
  lie on a (conjugate) diameter.  And its
  midpoint M is the centre of the ellipse.

The proof is trivial.  You just have to
consider what the figure looks like if
you project the ellipse (with the two
chords) into a circle.

You don't even need to know how to achieve
this projection.  You just need to know
that it can be done!

Very best wishes

Frank

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