Circle between quarter circles

The figure shows two quarter circles in a square and a circle that is tangent to the quarter circles and to a side of the square. The GREEN dot controls the size of the square. The BLUE dot controls the size of the circle. Can this figure be drawn with ruler & compass? If so, how? If not, why not? What questions could/would you put to your students based on this applet?
Inspired by Mind Your Decisions.