{{Short description|Circle constructed from an arbelos}} thumb|right|400px|A Bankoff circle with the center C'{{'}}<sub>6</sub>

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.<ref>{{citation|last=Bankoff|first=L.|authorlink=Leon Bankoff|title=Are the twin circles of Archimedes really twins?|journal=Mathematics Magazine|volume=47|pages=214–218|year=1974|issue=4 |doi=10.1080/0025570X.1974.11976399 |jstor=2689213}}.</ref><ref>{{citation|title=Those ubiquitous Archimedean circles|first1=Clayton W.|last1=Dodge|first2=Thomas|last2=Schoch|first3=Peter Y.|last3=Woo|first4=Paul|last4=Yiu|journal=Mathematics Magazine|volume=72|issue=3|year=1999|pages=202–213|doi=10.1080/0025570X.1999.11996731 |jstor=2690883}}.</ref><ref>{{citation|title=Configurations on centers of Bankoff circles |first=Zvonko |last=Čerin |journal=Far East Journal of Mathematical Sciences |volume=22 |issue=3 |year=2006 |pages=305–320 |url=http://web.math.unizg.hr/~cerin/c139.pdf |url-status=dead |archiveurl=https://web.archive.org/web/20110721102744/http://web.math.unizg.hr/~cerin/c139.pdf |archivedate=2011-07-21 }}.</ref>

==Construction== The Bankoff circle is formed from three semicircles that create an arbelos. A circle ''C''<sub>1</sub> is then formed tangent to each of the three semicircles, as an instance of the problem of Apollonius. Another circle ''C''<sub>2</sub> is then created, through three points: the two points of tangency of ''C''<sub>1</sub> with the smaller two semicircles, and the point where the two smaller semicircles are tangent to each other. ''C''<sub>2</sub> is the Bankoff circle.

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

:<math>R=\frac{1}{2}r\left(1-r\right).</math>

==References== {{reflist}}

== External links == * {{MathWorld|title=Bankoff Circle|urlname=BankoffCircle}} * [http://demonstrations.wolfram.com/BankoffCircle/ Bankoff Circle] by Jay Warendorff, the Wolfram Demonstrations Project. * [http://home.planet.nl/~lamoen/wiskunde/Arbelos/Catalogue.htm Online catalogue of Archimedean circles], Floor van Lamoen.

Category:Arbelos Category:Elementary geometry