The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). a. Given an arbelos in which the diameter of the largest circle is 1, what positions of point ?B ?maximize the area of the arbelos? b. Show that the area of the arbelos is the area of a circle whose diameter is the distance ?BD? in the figure.

Solution Step 1 : (a)The given diagram is Step 2: 1 Suppose the radius of the larger circle is r= 2 1 2 Area of the larger semi circle ,A = r 2 Radius of inner circle is given r (say) 1 Thus r =1 Then radius of second circle ,rr 1 Area of first inner circle with radius r is A = r 1 2 1 1 2 1 1 2 Area of first inner circle with radius r r is 1 = (r1r )2 1