Cyclic Quadrilateral Difference of Squares Theorem

The following result was experimentally discovered as a conjecture from a discussion that arose at the Stellenbosch Camp for talented South African Mathematics learners in 2007: If ΔABCD is a cyclic quadrilateral, E is the intersection of its diagonals, F and G are the respective circumcentre and incentre of ΔABE and H and I are the respective circumcentre and incentre of ΔDEC, then (DF2-CF2) - (DG2-CG2) = (AH2-BH2) - (AI2-BI2).

Drag any of the vertices A, B, C or D to dynamically move and change the figure to check your observation.

Cyclic Quadrilateral Difference of Squares Theorem

1) Can you explain why (prove that) the result is true?
2) Can you prove it in a different way?
3) Can you generalize the result further?

This problem appeared in the Problem Corner in the March 2015 issue of The Mathematical Gazette. Several readers provided insightful solutions, one of which is given in Solution-pp.167-168. These solutions showed that the result not only generalizes to any two similar triangles, but to any two pairs of similarly associated points in the two similar triangles. It is therefore a beautiful example of what has been called the 'discovery' function of proof, whereby proving a result leads to further generalization (see The role & function of proof in mathematics).
The problem has also been solved in different ways by Dirk Basson from South Africa, Waldemar Pompe from Poland, and Michael Fox from the UK.

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Created by Michael de Villiers, 9 July 2011; updated 7 Jan 2013; 27 June 2021.