Twin circles of Archimedes

Arbelos (black) and twin circles of Archimedes (blue)

The twin circles of Archimedes are two circles that are inscribed in an arbelos (also known as the sickle of Archimedes).

Definition and characteristics

If one draws the perpendicular to the diameter at the point of contact of the two inner semicircles in an arbelos, it divides this into two halves and in each half there is a circle that touches the outer semicircle, the corresponding inner semicircle and the vertical. These two circles are known as the twin circles of Archimedes because they are congruent . Their radius is: ${\displaystyle r}$

${\displaystyle r={\frac {r_{1}r_{2}}{r_{1}+r_{2}}}}$

Here and denote the radii of the two inner semicircles of the Arbelos. ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$

The common tangent of a twin circle and the associated inner semicircle goes through the end point of the diameter of the other inner semicircle. The smallest circle that the two twin circles touch from the inside is the same area as the Arbelos .:

Construction with compass and ruler

Construction of a twin circle

For a given arbelos, the three points on the base are designated by , and , so that the semicircle is above the outer semicircle of the arbelos and the semicircles above and its two inner semicircles. Furthermore, denote the intersection of the perpendicular in with the outer semicircle and the midpoint of the line . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle AC}$ ${\displaystyle AB}$ ${\displaystyle BC}$ ${\displaystyle D}$ ${\displaystyle B}$ ${\displaystyle M_{1}}$ ${\displaystyle AB}$

Now you construct the tangent from the point to the semicircle over . This touches the semicircle in and cuts the line in . Then you construct the bisector of the angle and the straight line , these intersect in the center of the twin circle, with the line as its radius. The second twin circle is obtained using a corresponding construction with the semicircle above . ${\displaystyle C}$ ${\displaystyle AB}$ ${\displaystyle T}$ ${\displaystyle BD}$ ${\displaystyle S}$ ${\displaystyle \angle DST}$ ${\displaystyle M_{1}T}$ ${\displaystyle U}$ ${\displaystyle UT}$ ${\displaystyle BC}$

Historical

The construction can be found in the Book of Lemmas , whose attribution to Archimedes is questionable.

See also

• Archimedean Circle
• Bankoff circles
• Apollonian problem - the general case of constructing a circle of contact with three given circles, lines or points

literature

• Günter Aumann : Circular Geometry: An Elementary Introduction . Springer, 2015, ISBN 9783662453063 , pp. 193-200
• Leon Bankoff: Are the Twin Circles of Archimedes Really Twins? . Mathematics Magazine, Vol. 47, No. 4 (Sep., 1974), pp. 214-218 ( JSTOR )
• Clayton W. Dodge, Thomas Schoch, Peter Y. Woo, Paul Yiu: Those Ubiquitous Archimedean Circles . Mathematics Magazine, Vol. 72, No. 3 (Jun., 1999), pp. 202-213 ( JSTOR )
• Shailesh A. Shirali: A generalization of the arbelos theorem of Archimedes . The Mathematical Gazette, Vol. 95, No. 533 (July 2011), pp. 197-205 ( JSTOR )

Web links

Commons : Archimedes' twin circles  - collection of images, videos and audio files

• Animated proof of the twin circles of Archimedes , State Education Server Baden-Württemberg
• Eric Weisstein Archimedes Circles , Wolfram Mathworld
• Interactive diagram that visualizes numerous Archimedes' circles

Individual evidence

1. ↑ ^{a b c} Günter Aumann: Circle geometry: An elementary introduction . Springer, 2015, ISBN 9783662453063 , pp. 193-200