Start with any cyclic quadrilateral. Draw one of the diagonals and inscribe red circles in the resulting triangles. Then repeat for the other diagonal, creating two blue circles.
The Japanese theorem for quadrilaterals: The sum of the radii of the red circles equals the sum of the radii of the blue circles.
(Neat fact: the centers of the four circles form a rectangle.)
The circle shown below has radius 1. Slide the points around the circle without changing their order.