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@wence
I have that the red area is 9/4 pi + 1 - 2 sqrt(2) - 9 arcsin(1/3), and the sum of the two blue pieces (assuming they don't overlap with the red piece at all) is 9 arcsin(1/3) + 2 sqrt(2) - 2.
arcsin 1/3 is not a "nice" number I think, so I don't see a natural easy ratio