Thermal capillary wave

The Archimedes' circles (red) have the same area. The large semicircle has unit diameter, BC = 1–r, and AB = r = AB/AC.

In geometry, Archimedes' circles, first created by Archimedes, are two circles that can be created inside of an arbelos, both having the same area as each other.

Construction

From any three collinear points A, B, and C, one may form an arbelos, a shape bounded by three semicircles having pairs of these three points as their diameters. All three semicircles must be on the same side of line AC. Archimedes' twin circles are created by drawing a perpendicular line to line AC through the middle point B of the three given points, tangent to the two smaller semicircles. Each of the two circles C₁ and C₂ is tangent to that line and to the large semicircle; C₁ is tangent to one of the smaller semicircles and C₂ is tangent to the other smaller semicircle. Each of the two circles is uniquely determined by its three tangencies; constructing each of the twin circles from its tangencies is a special case of the Problem of Apollonius.

Radii of the circles

Because the two circles are congruent, they both share the same radius length. If r = AB/AC, then the radius of either circle is:

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

Also, according to Proposition 5 of Archimedes' Book of Lemmas, the common radius of any Archimedean circle is:

$\rho ={\frac {ab}{a+b}}$

where a and b are the radii of two inner semicircles.

Centers of the circles

If r = AB/AC, then the centers to C₁ and C₂ are:

$C_{1}=\left({\frac {1}{2}}r\left(1+r\right),r{\sqrt {1-r}}\right)$

$C_{2}=\left({\frac {1}{2}}r\left(3-r\right),\left(1-r\right){\sqrt {r}}\right)$

See also

• Bankoff circle
• Schoch circles
• Schoch line
• Woo circles

References

• Template:Citeweb

External links

• A catalog of over fifty Archimedean circles