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