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.
Cf. [a1] for a higher-dimensional generalization.
|[a1]||K. Borsuk, "Analytic geometry" , PWN (1969)|
Apollonius theorem. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Apollonius_theorem&oldid=17516