The large circle in the diagram has radius 1. Circles centered at A and B are tangent and each tangent to the large circle. (The shape in between the circles is called an arbelos after the Greek for a cobbler’s knife, which it apparently resembles.)

BC is tangent to circle A and AD tangent to circle B.

EF and GH are perpendicular to AB.

Can you show that they are equal?

Can you find an expression for their length in terms of the radius of circle A?