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Topic: An interesting property of ellipses
Replies: 1   Last Post: Jun 23, 1994 12:21 PM

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 Rouben Rostamian Posts: 4 Registered: 12/6/04
An interesting property of ellipses
Posted: Jun 22, 1994 9:27 PM

I recently stumbled upon an interesting property of ellipses that I
had not seen before. I can prove it with not much trouble by analytic
geometry. But unfortunately my proof does not offer much in terms of
insight. Can anyone come up with an "insightful" demonstration of why
this is true?

Proposition:
The length of the diagonal of a circumscribing rectangle
of an ellipse is independent of the rectangle's orientation.

Corollary:
The locus of vertices of all circumscribing rectangles of an
ellipse is a circle.

Note 1: A "circumscribing rectangle of an ellipse" is a
rectangle with all its four sides tangent to the ellipse.
(The sides of the rectangle need not be parallel with the
axes of the ellipse.)

Note 2: If the major and minor semiaxes of the ellipse have
lengths a and b, it can be shown that the length of the diagonal
of the circumscribing rectangle is twice sqrt(a^2+b^2).

--
Rouben Rostamian <rouben@math.umbc.edu>

Date Subject Author
6/22/94 Rouben Rostamian
6/23/94 John Conway