The following conjecture was discovered a few months ago, but I've not had any time yet to work on it: 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 Conjecture
This problem (with solutions later) is scheduled to appear in the Problem Corner in a future issue of The Mathematical Gazette. So far the problem has 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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Michael de Villiers, 9 July 2011. Updated 7 Jan 2013.