Four semicircles are arranged around the edges of a rectangle. What are their respective radii?

This is a great deal more difficult than ‘Three Semicircles in a Square’. If it helps I can tell you that there is exactly one solution, and that each of the radii are rational numbers (so if you find it useful – and I’m not at all sure you will - you can scale the whole thing up by some factor, solve as a Diophantine set of equations, and scale back down again). I myself found this very difficult to solve, so if you discover a route to the solution that is not too difficult, I would be interested in knowing it.