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.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

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

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

71921606