Between the seventeenth and nineteenth centuries, Japan was completely isolated from the West. During that period, the Japanese developed their own unique style of mathematics. People from all walks of life would inscribe geometry problems on wooden tablets called sangaku (in color) and hang them in temples throughout Japan.
This month's problem is a sangaku problem: Two quarter-circles and a semi-circle are inscribed in a unit square as shown in the figure below. The smaller green circle is tangent to the two quarter-circles and to the side of the square. The larger green circle is tangent to the two quarter-circles and to the semi-circle. Find the radii of the two green circles.