Japanese theorem for cyclic quadrilaterals

In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.

It was originally stated on a sangaku tablet on a temple in Yamagata prefecture, Japan, in 1880.

Specifically, let □ABCD be an arbitrary cyclic quadrilateral and let M1, M2, M3, M4 be the incenters of the triangles △ABD, △ABC, △BCD, △ACD.

[1] This theorem may be extended to prove the Japanese theorem for cyclic polygons, according to which the sum of inradii of a triangulated cyclic polygon does not depend on how it is triangulated.

[2] The quadrilateral case immediately proves the general case, as any two triangulations of an arbitrary cyclic polygon can be connected by a sequence of flips that change one diagonal to another, replacing two incircles in a quadrilateral by the other two incircles with equal sum of radii.