Bankoff circle

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]

A Bankoff circle with the center C''₆

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:

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