7. 

6. In the Book of Lemmas (a collection of 13 geometrical propositions that has come down to us only in an Arabic translation), Archimedes introduced a gure that, owing to its shape, is known as the arbelos, or "shoemaker's knife." A C B If a straight line AB is divided into two parts at C and if on one side of AB are described semicircles with AB, AC, and CB as diameters, then the region included between the circumferences of the three semicircles is the shoemaker's knife. Prove that if PC is the straight line perpendicular to AB at C, then the area of the shoemaker's knife equals the area of the circle whose diameter is PC. [Hint: AB? = AC + BC + 2AC BC = AC + BC +... +2PC] 7. Prove that if the common external tangent to the two smaller semicircles in the shoemaker's knife touches these curves at R and S, then RS and PC bisect each other, and R, S, P, and C lie on the circle whose diameter is PC.