Conjecture
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).
Explore
1) 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
Challenge
2) Can you explain why (prove that) the result is true?
3) Can you prove it in a different way?
4) Can you generalize the result further?
Solutions
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 Problem Corner Solution-pp.167-168. The problem was also solved in different ways by Dirk Basson from South Africa, Waldemar Pompe from Poland, and Michael Fox from the UK. Links to their solutions are give below:
Solution by Michael Fox
Solution by Dirk Basson
Solution by Waldemar Pompe
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 another 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).
Some Related Links
Cyclic Hexagon Alternate Angles Sum Theorem
Circumscribed Hexagon Alternate Sides Theorem
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Angle Divider Theorem for a Cyclic Quadrilateral
Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Triangle Circumcircle Incentre Result
Cyclic Kepler Quadrilateral Conjectures
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
Japanese theorem for cyclic quadrilaterals
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Tangential Quadrilateral Theorem of Gusić & Mladinić
Bradley's Theorem, its Generalization & an Analogue Theorem
Euler-Nagel line Duality (analogy)
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
Matric Exam Geometry Problem - 1949 (A variation of Reim's theorem)
International Mathematical Talent Search (IMTS) Problem Generalized
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Similar Parallelograms: A Generalization of a Golden Rectangle property
Eight Point Conic for Cyclic Quadrilateral
Geometry Loci Doodling with Cyclic Quadrilaterals
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
An Inclusive, Hierarchical Classification of Quadrilaterals
Some External Links
Cyclic quadrilateral (Wikipedia)
Incenter (Wikipedia)
Circumcircle (Wikipedia)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers as well as books on problem solving can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Created by Michael de Villiers, 9 July 2011 with JavaAketchpad; updated 7 Jan 2013; Updated to WebSketchpad 27 June 2021; 18 May 2025.