# Archimedean circle

Archimedes' twin circles. The large semicircle has unit diameter, BC = 1–r, and AB = r = AB/AC.

In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. The radius ρ of such a circle is given by

${\displaystyle \rho ={\frac {1}{2}}r\left(1-r\right),}$

where r is the ratio AB/AC shown in the figure to the right. There are over fifty different known ways to construct Archimedean circles.[1]

## Origin

An Archimedean circle was first constructed by Archimedes in his Book of Lemmas. In his book, he constructed what is now known as Archimedes' twin circles.

## Other Archimedean circles finders

### Leon Bankoff

Leon Bankoff has constructed other Archimedean circles called Bankoff's triplet circle and Bankoff's quadruplet circle.

The Schoch line (cyan line) and examples of Woo circles (green).

### Thomas Schoch

In 1978 Thomas Schoch found a dozen more Archimedean circles (the Schoch circles) that have been published in 1998.[2][3] He also constructed what is known as the Schoch line.[4]

### Peter Y. Woo

Peter Y. Woo considered the Schoch line, and with it, he was able to create a family of infinitely many Archimedean circles known as the Woo circles.[5]

### Frank Power

In the summer of 1998, Frank Power introduced four more Archimedes circles known as Archimedes' quadruplets.[6]

## References

1. ^ "Online catalogue of Archimedean circles". Retrieved 2008-08-26.
2. ^ Thomas Schoch (1998). "A Dozen More Arbelos Twins". Retrieved 2008-08-30.
3. ^ Clayton W. Dodge; Thomas Schoch; Peter Y. Woo; Paul Yiu (1999). "Those Ubiquitous Archimedean Circles" (PDF). Retrieved 2008-08-30.
4. ^ van Lamoen, Floor. "Schoch Line." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein". Retrieved 2008-08-26.
5. ^ Thomas Schoch (2007). "Arbelos - The Woo Circles". Retrieved 2008-08-26.
6. ^ Power, Frank (2005). "Some More Archimedean Circles in the Arbelos". In Yiu, Paul. Forum Geometricorum. 5 (published 2005-11-02). pp. 133–134. ISSN 1534-1178. Retrieved 2008-06-26.