If a parallelogram is circumscribing an ellipse, the point of contacts are the vertices of a parallelogram. How can one prove that the diagonals of the first parallelogram are parallel to the sides of the second parallelogram?

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- Dec 15th 2008, 07:03 AMgeo_mathparallelogram
If a parallelogram is circumscribing an ellipse, the point of contacts are the vertices of a parallelogram. How can one prove that the diagonals of the first parallelogram are parallel to the sides of the second parallelogram?

- Dec 18th 2008, 08:26 PMkalagota
it would be better if you could give an illustration..

- Dec 19th 2008, 12:13 AMOpalg
This result is more or less obvious if the ellipse happens to be a circle. The circumscribing parallelogram will be a rhombus, and the second parallelogram will be a rectangle formed by joining the points of contact of the rhombus with the circle.

But you can get any ellipse from a circle by a contraction in the direction of the minor axis. Such a contraction takes straight lines to straight lines and preserves parallelism (though it does not preserve lengths or angles). So it will take the circumscribing rhombus to a parallelogram, and the inner rectangle to a parallelogram whose sides are parallel to the diagonals of the outer parallelogram. Any circumscribing parallelogram arises in this way.