Circle squared

Calculus Level 3

Consider a square ABCDABCD. Two circles XX and Y Y are drawn such that circle XX is tangent to two adjacent sides ABAB and ADAD, and circle YY is tangent to the other two adjacent sides CBCB and CD,CD, and circles XX and YY are tangent to each other.

As the radius of circle XX increases, at what radius of circle XX does the sum of the areas of circle XX and YY change the slowest?

Express that radius as a fraction of the side of the square ABCDABCD, and round to the nearest hundredth.

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