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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 in 1974.[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.

A Bankoff circle with the center C''6

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

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

References

Bankoff, L. (1974), "Are the twin circles of Archimedes really twins?", Mathematics Magazine, 47: 214–218, JSTOR 2689213.
Dodge, Clayton W.; Schoch, Thomas; Woo, Peter Y.; Yiu, Paul (1999), "Those ubiquitous Archimedean circles", Mathematics Magazine, 72 (3): 202–213, JSTOR 2690883.

Čerin, Zvonko (2006), "Configurations on centers of Bankoff circles" (PDF), Far East Journal of Mathematical Sciences, 22 (3): 305–320, archived from the original (PDF) on 2011-07-21.

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