By: Tao Steven Zheng (鄭濤)

【Problem】

In the "Book of Lemmas", Thabit ibn Qurra (c. 826 – 901 AD) describes a shape called the arbelos. The arbelos in the figure below is the shape bounded by three semicircles with diameters AB, AD, and BD (shaded in blue). Suppose a circle be constructed such that its diameter is CD (shaded in red), determine the ratio of the areas of the arbelos to the circle.

[1] It is uncertain whether the "Book of Lemmas" was authored by Thabit ibn Qurrah or Archimedes because the earliest edition is written in Arabic and refers to Archimedes in the third person.

【Solution】

Area of the circle

Area of the arbelos

Since

Therefore,

and