Two squares share a vertex. Prove that the red point is the circumcentre of the red triangle.
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Stay centred
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Quadrilateral AFEB is cyclic because angle A and E are 90. Hence, point O is circumcenter of cyclic quadrilateral ABEF. It means point O lies on perpendicular bisector AB and EF, hence it also lies on perpendicular bisector of CD and DG (since ABCD and DEFG are square).
Hence it should be circumcenter of triangle CDG.