Archimedean Circle

Twin circles of Archimedes. The large semicircle has the diameter 1, BC = 1- r , and AB = r = AB / AC

In geometry , an Archimedes' circle is a circle that can be constructed with the help of an Arbelos and that is congruent to the twin circles of Archimedes . If one normalizes the Arbelos so that the diameter of the outer (largest) semicircle is 1, and denotes the radius of one of the two smaller semicircles , the result is the radius of an Archimedean circle ${\displaystyle r}$ ${\displaystyle \rho }$

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

More than 60 different construction possibilities of Archimedean circles are known.

The first constructions of Archimedean circles are the twin circles constructed in the Book of Lemmas , ascribed to the Greek mathematician Archimedes .

Examples of Archimedean circles

Bankoff triplet circle (blue)		Bankoff quadruplet circle (blue)

Schoch-Straight (cyan) and Schoch-Kreis W15		Schoch circle W15 (light green) and example of a woo circle (above)

Power circles

Bankoff circles

The American dentist and mathematician Leon Bankoff discovered the Bankoff circles named after him in 1954 and 1974 . Since these were historically the third and fourth of the Archimedean circles after the Archimedes' twins, they are also called Bankoff triplet circle (in German: "Bankoffs Drillings-Kreis") and Bankoff quadruplet circle ("Bankoffs Vierlings-Kreis") .

Schoch circles and Schoch straight lines

In 1978 the German Thomas Schoch discovered a dozen more Archimedean circles, the so-called Schoch circles , which were published in 1998 . He also constructed the Schoch straight line . This is constructed with the help of two further circles with a center or ( and ) and the largest semicircle of the Arbelos ( ). The circle with the center point is constructed tangential to these arcs . The perpendicular straight through to is the Schoch straight. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$ ${\displaystyle K_{3}}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{1}}$ ${\displaystyle AB}$

Woo circles

With the help of the Schoch line, Peter Y. Woo succeeded in finding a family of an infinite number of Archimedean circles, the so-called Woo circles . He showed: is a positive real number and if two tangent circles with a center on the base line of the Arbelos and the -fold radius of the two smaller Arbelos circles are constructed (in the figure the red and the blue circle), then it is tangential Circle lying to these two circles with center on the Schoch line congruent to the Archimedean twin circles, thus an Archimedean circle. ${\displaystyle m}$ ${\displaystyle C}$ ${\displaystyle m}$

Power circles

In the summer of 1998, Frank Power presented four more Archimedes 'circles, the so-called power circles , which are also known as Archimedes' quadruplets in English . They are constructed as follows: If and are the radii of the two small Arbelos circles, the center of the semicircle with radius , the point perpendicular to above on the circle, and the center of the line , then the two are in tangent circles, which also have the Touch the outer Arbelos circle, two of the four power circles. The other two power circles are constructed analogously with the semicircle with a radius . ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$ ${\displaystyle D}$ ${\displaystyle r_{1}}$ ${\displaystyle E}$ ${\displaystyle AC}$ ${\displaystyle D}$ ${\displaystyle H}$ ${\displaystyle AC}$ ${\displaystyle HE}$ ${\displaystyle E}$ ${\displaystyle r_{2}}$

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1. ↑ Online catalog of Archimedean circles. home.wxs.nl, accessed on December 14, 2015 . 
2. ↑ Thomas Schoch: A Dozen More Arbelos Twins. In: retas.de. Biola University , January 1998, accessed December 14, 2015 . 
3. ^ Clayton Dodge, Thomas Schoch, Peter Woo, Paul Yiu: Those Ubiquitous Archimedean Circles. (PDF; 895 KB) In: retas.de. Biola University, June 1999, accessed December 14, 2015 . 
4. ^ Floor van Lamoen: Schoch Line . In: MathWorld (English).
5. ↑ Thomas Schoch: Arbelos - The Woo Circles. In: retas.de. Biola University, 2007, archived from the original on August 14, 2014 ; accessed on December 14, 2015 . 
6. ^ Frank Power: Some More Archimedean Circles in the Arbelos. (PS; 112 KB) In: forumgeom.fau.edu. Florida Atlantic University , November 2, 2005, accessed December 14, 2015 .