Colored Tiles And Tangent Circles

Geometry Level 3

As shown in the figure, four circles are inscribed within a square. Each circle is tangent to both of its adjacent circles and the midpoint of the square's nearest side.

By connecting the midpoints for each of the square's sides, one creates a light blue square. Likewise, by drawing four line segments between each of the circles' centers, one creates a pink square. Find the ratio of the area enclosed by the larger blue square and the area enclosed by the smaller pink square. If this proportion can be expressed in the form \(\frac { A }{ B } +\sqrt { C }\), with \(A\), \(B\), and \(C\) primes, find \(A+B+C\).

Details and Assumptions:

The area enclosed by the blue square includes the pink square as well.

This problem is original. The picture was produced from Desmos and MS Paint.

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