The Arbelos 4: The Inscribed Circle

Topic:
Circle
Thanks http://tube.geogebra.org/stevephelps for sharing this fantastic applet! The figure below is an Arbelos, which means shoemaker's knife in Greek. The grey circle is the Inscribed Circle of the Arbelos. Drag the blue points. The red circles suggest a way to construct the inscribed circle. How should the inscribed circle be constructed?