What is an arbelos?
Properties of an arbelos
        Distances and areas
        Archimedes' circles
        Apollonius circle
        Bankoff circle
        Pappus chain

Stephen Tan
Dec 16, 2005
Math 308 by Dr. Bill Casselman
Final Project
Page 1
Label the diameters of the left and right semicircles r and 1 - r, so that the enclosing semicircle has diameter 1. The arc length along the bottom of the arbelos is:

The arc length along the two smaller semicircles is the same as the arc length of the enclosing semicircle.

Archimedes' Book of Lemmas, Proposition IV
Let D be any point on a semicircle of diameter AC, and let BD be perpendicular to AC. If semicircles be described within the first semicircle and having AB and BC as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called arbelos". Then the area of the arbelos is the same as the area of the circle with diameter BD.
Page 2
Let B be the point at the notch of the arbelos. By drawing a perpendicular line BD to the circumference of the enclosing circle, the area of the circle with the diameter BD is equal to the area of the arbelos.