Bankoff circle

A Bankoff circle with the center C''₆

In geometry, the Bankoff circle or Bankoff triplet circle is a certain Archimedean circle that can be constructed from an arbelos; an Archimedean circle is any circle with area equal to each of Archimedes' twin circles. The Bankoff circle was first constructed by Leon Bankoff.^[1][2][3]

Construction

The Bankoff circle is formed from three semicircles that create an arbelos. A circle C₁ is then formed tangent to each of the three semicircles, as an instance of the problem of Apollonius. Another circle C₂ is then created, through three points: the two points of tangency of C₁ with the smaller two semicircles, and the point where the two smaller semicircles are tangent to each other. C₂ is the Bankoff circle.

Radius of the circle

If r = AB/AC, then the radius of the Bankoff circle is:

$R=\frac{1}{2}r\left(1-r\right).$

References

1. Bankoff, L. (1974), "Are the twin circles of Archimedes really twins?", Mathematics Magazine 47: 214–218, JSTOR 2689213.
2. Dodge, Clayton W.; Schoch, Thomas; Woo, Peter Y.; Yiu, Paul (1999), "Those ubiquitous Archimedean circles", Mathematics Magazine 72 (3): 202–213, JSTOR 2690883.
3. Čerin, Zvonko (2006), "Configurations on centers of Bankoff circles" (PDF), Far East Journal of Mathematical Sciences 22 (3): 305–320.

External links

• Weisstein, Eric W., "Bankoff Circle", MathWorld.
• Bankoff Circle by Jay Warendorff, the Wolfram Demonstrations Project.
• Online catalogue of Archimedean circles, Floor van Lamoen.