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.[br][br]The GREEN dot controls the size of the square. The BLUE dot controls the size of the circle.[br][br]Can this figure be drawn with ruler & compass? If so, how? If not, why not?[br][br][color=#ff0000][i][b]What questions could/would you put to your students based on this applet?[/b][/i][/color]
Inspired by Mind Your Decisions.

Information: Circle between quarter circles