The Area of Arbelos

An arbelos is a shape enclosed by three semicircles with collinear centers, to which Archimedes has devoted several propositions to in his Book of Lemmas.

Below is an attempt to present a proof without words due to R. Nelsen of Proposition 4 from the Book of Lemmas. I made a valiant effort, but the result does not seem as clear as the original pencil and paper diagram. The proof is based on a triple application of Euclid's proof of the Pythagorean theorem.

Three blue and three red semicircles account for two applications of Euclid's proof. The third one combines the big semicircle, the biggest of the blue and the biggest of the red ones.

References

1. R. Nelsen, Proof Without Words: The Area of Arbelos, Mathematics Magazine, 75 (2002), p. 144.

• Arbelos - the Shoemaker's Knife
• 7 = 2 + 5 Sangaku
• Another Pair of Twins in Arbelos
• Archimedes' Quadruplets
• Archimedes' Twin Circles and a Brother
• Book of Lemmas: Proposition 5
• Book of Lemmas: Proposition 6
• Chain of Inscribed Circles
• Concurrency in Arbelos
• Concyclic Points in Arbelos
• Ellipse in Arbelos
• Gothic Arc
• Pappus Sangaku
• Rectangle in Arbelos
• Squares in Arbelos
• The Area of Arbelos
• Twin Segments in Arbelos
• Two Arbelos, Two Chains

Copyright © 1996-2008 Alexander Bogomolny