Archimedes' twin circles

In geometry, Archimedes' circles, first created by Archimedes, are two circles that can be created inside of an arbelos with the same area.

Construction

The Archimedes' circles are created by taking three semicircles to form an arbelos. A perpendicular line to line AC is then made from the intersection of the two smaller semicircles. The two circles "C"₁ and "C"₂ are both tangent to that line, the large semicircle, and one each of the smaller semicircles.

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:

:$ho=frac\{1\}\{2\}rleft(1-r\; ight)$

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

:$ho=frac\{ab\}\{a+b\}$

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

Centers of the circles

If "r" = "AB"/"AC", then the centers to "C"₁ and "C"₂ are:

:$C\_1=left(frac\{1\}\{2\}rleft(1+r\; ight),rsqrt\{1-r\}\; ight)$:$C\_2=left(frac\{1\}\{2\}rleft(3-r\; ight),left(1-r\; ight)sqrt\{r\}\; ight)$

ee also

* Schoch line

References

*citeweb|author=Weisstein, Eric W|title="Archimedes' Circles." From MathWorld--A Wolfram Web Resource|url=http://mathworld.wolfram.com/ArchimedesCircles.html|accessdate=2008-04-10

External links

* [http://home.planet.nl/~lamoen/wiskunde/Arbelos/Catalogue.htm A catalog of over fifty Archimedean circles]