Bankoff circles

Bankoff triplet circle (blue)

Bankoff quadruplet circle (blue)

The Bankoff circles are two circles in geometry that can be constructed in an Arbelos and that have the same radius as the two twin circles of Archimedes . They belong to the so-called Archimedean circles . The Bankoff circles are named after the American dentist and mathematician Leon Bankoff , who discovered them in 1954 and 1974 .

Since the Bankoff circles - after the "Archimedean Twins" - historically the third and fourth of the (as we know today) infinitely many Archimedean circles, they are also called Bankoff triplet circle (in German, for example: "Bankoffs Drillings-Kreis ") And Bankoff quadruplet circle (" Bankoffs Vierlings-Kreis "). The German names are not used, however, so the English ones are used here.

construction

An arbelos is formed by the three semicircles above , and (black in the drawings). The twin circles of Archimedes are drawn in light gray in the illustrations. ${\displaystyle AB}$ ${\displaystyle BC}$ ${\displaystyle AC}$

Bankoff triplet circle

(top picture)

Draw the inscribed circle of Arbelos (orange), i.e. the circle that, according to the problem of Apollonius, touches the three semicircles of Arbelos . The circle (blue) that goes through and the two points of contact of the inscribed circle with the smaller Arbelos semicircles is the Bankoff triplet circle . ${\displaystyle B}$

Numerous other, sometimes astonishing, properties of this Bankoff circle were discovered in the 2000s by the Dutchman Floor van Lamoen and others and documented by him in his "Online catalog of Archimedean circles".

Bankoff quadruplet circle

(lower figure)

Draw the common tangent (orange) of the two smaller Arbelos semicircles that do not go through . The largest circle (blue) in the area between this tangent and the large Arbelos arc is the Bankoff quadruplet circle . It touches the large Arbelos semicircle at the point where the vertical line established on the Arbelos baseline intersects the large Arbelos arch. ${\displaystyle B}$ ${\displaystyle D}$ ${\displaystyle B}$ ${\displaystyle AC}$

Bankoff circle radius

If the radii of the two smaller Arbelos semicircles are denoted by or , then the following applies to the radius of each of the two Bankoff circles: ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$ ${\displaystyle R}$

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

The Bankoff circles thus have the same radius as the twin circles of Archimedes (light gray in the drawings).

See also

• Arbelos
• Twin circles of Archimedes

Web links

Commons : Bankoff circles  - collection of images

• Floor van Lamoen: Online catalog of Archimedean circles . (English)
• Eric W. Weisstein : Bankoff Circle . In: MathWorld (English).
• Jay Warendorff: Bankoff Circle . In: Wolfram Demonstrations Project (English)
• Clayton W. Dodge, Thomas Schoch, Peter Y. Woo, Paul Yiu: Those Ubiquitous Archimedean Circles (PDF; 916 kB; English)

Individual evidence

1. ↑ Leon Bankoff: Are the twin circles of Archimedes really twins? In: Mathematics Magazine , MAA 1974, Vol. 47, No. 4, pp. 214-218.
2. ^ Clayton W. Dodge, Thomas Schoch, Peter Y. Woo, Paul Yiu: Those Ubiquitous Archimedean Circles . In: Mathematics Magazine , MAA 1999, No. 72, pp. 202–213 (for facsimile see web links).
3. ^ Floor van Lamoen: Bankoff's Triplet circle. Retrieved March 20, 2012 .