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Apollonius theorem

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The sum of the squares of the lengths of conjugate semi-diameters of an ellipse is constant and equal to the sum of the squares of the lengths of its semi-axes.

The area of a parallelogram circumscribed around an ellipse, with its sides oriented in opposite directions, is constant and equal to the product of the lengths of its diameters.


Comments

Cf. [a1] for a higher-dimensional generalization.

References

[a1] K. Borsuk, "Analytic geometry" , PWN (1969)
How to Cite This Entry:
Apollonius theorem. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Apollonius_theorem&oldid=17516
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098