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).

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.