We've got you cornered .....Geometry Level 5
Suppose that two circles of equal radius are drawn within a square such that each is centered on the same diagonal of the square and each is tangent to precisely two sides of the square, as well as to one another at precisely one point.
Next, at each of the two corners not on this diagonal are centered quarter-circles of equal radius, such that they are both tangent at precisely one point to each of the two circles previously drawn.
Let \(S\) be the ratio of the combined areas of the two circles and two quarter-circles to the area of the square. Find \(\lfloor 10000*S \rfloor\).