Gregory Ellipse Quadrilateral Generalization

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Allaire, Zhou and Yao (2012) give a proof of a result apparently first stated by Gregory in the 19th century that, if a triangle ABC circumscribes an ellipse with foci P and Q, then PA.AQ/CA.AB + PB.BQ/AB.BC + PC.CQ/BC.CA = 1.

This result generalizes to a convex quadrilateral ABCD circumscribed around an ellipse, then PA.AQ/DA.AB + PB.BQ/AB.BC + PC.CQ/BC.CD + PD.DQ/CD.DA = 2, as shown in De Villiers & Fox (2014).

1) P. R. Allaire, J. Zhou & H. Yao, Proving a nineteenth century ellipse identity, Math. Gaz. 96 (March 2012), pp.161 – 165.
2) M. de Villiers & M. Fox, Generalisations of a 19th century result on ellipses, Math. Gaz. (Nov 2014), pp. 414-423.

Gregory Ellipse Quadrilateral Generalization

Download our paper at the link given in the References above.

A possibly suitable challenge for mathematically talented high school learners might be to ask them to explore the special case when the ellipse is a circle, and the quadrilateral becomes a tangential (circumscribed) quadrilateral.

I am grateful to Taxia Limneou from Greece for recently (8 Sept 2021) posting the following straightforward, elementary proof, as well as some corollaries, of the above Gregory generalization, at the 'Romantics of Geometry' Facebook group: Taxia's proof using Poncelet.

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Created by Michael de Villiers, 3 December 2013. Updated 21 Feb 2015; 17 August 2020.