## Cyclic Quadrilateral Conjecture

The following conjecture was discovered a few months ago, but I've not had any time yet to work on it: If Δ*ABCD* is a cyclic quadrilateral, *E* is the intersection of its diagonals, *F* and *G* are the respective circumcentre and incentre of Δ*ABE* and *H* and *I* are the respective circumcentre and incentre of Δ*DEC*, then (*DF*^{2}-*CF*^{2}) - (*DG*^{2}-*CG*^{2}) = (*AH*^{2}-*BH*^{2}) - (*AI*^{2}-*BI*^{2}).

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

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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.