In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.[1]: p.
193 Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic.
This theorem can be proven by first proving a special case: no matter how one triangulates a cyclic quadrilateral, the sum of inradii of triangles is constant.
The steps of this theorem require nothing beyond basic constructive Euclidean geometry.
[2] With the additional construction of a parallelogram having sides parallel to the diagonals, and tangent to the corners of the rectangle of incenters, the quadrilateral case of the cyclic polygon theorem can be proved in a few steps.