Japanese theorem for concyclic quadrilaterals
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The Japanese theorem states that the centers of the incircles of certain triangles inside a concyclic quadrilateral are vertices of a rectangle.
Triangulate an arbitrary concyclic quadrilateral by its diagonals, this yields four overlapping triangles (each diagonal creates two triangles). The centers of the incircles of those triangles form a rectangle.
Let 
be an arbitrary concyclic quadrilateral and let be I,M,G,K the incenters of the triangles 
then the quadrilateral formed by I,M,G,K is a rectangle.
Note that this theorem is easily extended to prove the Japanese theorem for concyclic polygons. To prove the quadrilateral case, simply construct the parallelogram tangent to the corners of the constructed rectangle, with sides parallel to the diagonals of the quadrilateral. The construction shows that the parallelogram is a rhombus, which is equivalent to showing that the sums of the radii of the incircles tangent to each diagonal are equal.
The quadrilateral case immediately proves the general case by induction on the set of triangulating partitions of a general polygon.
