# Archimedean circle

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]

### Archimedean circles in Wasan geometry (Japanese geometry)

In 1831, Nagata (永田岩三郎遵道) proposed a sangaku problem involving two Archimedean circles, which are denoted by W6 and W7 in [3]. In 1853, Ootoba (大鳥羽源吉守敬) proposed a sangaku problem involving an Archimedean circle. [7]

## 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". Archived from the original on 2014-08-14. Retrieved 2008-08-26.
6. Power, Frank (2005). "Some More Archimedean Circles in the Arbelos". In Yiu, Paul (ed.). Forum Geometricorum. 5 (published 2005-11-02). pp. 133–134. ISSN 1534-1178. Retrieved 2008-06-26.
7. Okumura, Hiroshi (2019). "Remarks on Archimedean circles of Nagata and Ootoba". In Okumura, Hiroshi (ed.). Sangaku Journal of Mathematics (PDF). 3 (published 2019-11-04). pp. 119–122. ISSN 2534-9562. Retrieved 2019-11-04.