Japanese theorem for cyclic quadrilaterals
Triangulating an arbitrary concyclic quadrilateral by its diagonals yields four overlapping triangles (each diagonal creates two triangles). The centers of the incircles of those triangles form a rectangle.
Specifically, let be an arbitrary concyclic quadrilateral and let be the incenters of the triangles . Then the quadrilateral formed by is a rectangle.
Note that this theorem is easily extended to prove the Japanese theorem for cyclic 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.
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