# Apollonius theorem

From Encyclopedia of Mathematics

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