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Osječki matematički list, Vol. 20 No. 1, 2020.

Professional paper

Apolonius’ theorems

Predrag Lončar orcid id orcid.org/0000-0003-1823-7682 ; Geotehnički fakultet, Varaždin

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Abstract

In this paper, we define the conjugate diameters of an ellipse and consider their properties. We prove two important relations holding for conjugate diameters of an ellipse, i.e. the famous theorems of Apollonius. According to the first Apollonius’ theorem, the sum of squares of the lengths of any two conjugate radii of the ellipse is equal to the sum of squares of the lengths of its semi -major and semi- minor axes. The second Apollonius’ theorem claims that the area of the parallelogram determined by any pair of conjugate diameters of an ellipse is equal to the area of a tangential rectangle defined by the major and minor axes of this ellipse. Some interesting statements are made and proved by means of Apollonius’ theorems.

Keywords

ellipse, diameter of ellipse, conjugate diameters of ellipse, tangential parallelogram of ellipse, Apollonius’ theorems

Hrčak ID:

251788

URI

https://hrcak.srce.hr/251788

Publication date:

15.6.2020.

Article data in other languages: croatian

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