Arbelos

In Archimedes' Book of Lemmas (ca 250), he introduces a figure that, due to its shape, has historically been known as "the shoemaker's knife" or arbelos. If, in a given semicircle with radius R and diameter AB, two semicircles with radii r₁ and r_2, where r₁ does not equal r₂ and r₁ + r₂ = R, are constructed on diameter AB so that they meet at point C on AB, then the region bounded by the three circumferences is called an arbelos.

The arbelos fascinated Archimedes with its mathematical properties. Let us explore some of these properties:

a) Prove that the length of arc AC plus the length of arc CB equals the length of arc AB

b)Prove: If a perpendicular line is constructed from C intersecting the arc AB at point P, then PC is the diameter of a circle whose area equals that of the arbelos.

c) Complete the lower half of the circle with diameter AB, let the midpoint of the arc of this lower semicircle be Q, the midpoint of arc AC be M and the midpoint of arc CB be N. Prove that the area of quadrilateral MCNQ is equal to r₁^2₊ r₂^2.

A C B