Archimedes' Quadruplets

One of the properties of the arbelos noticed and proved by Archimedes in his Book of Lemmas is that the two small circles inscribed into two pieces of the arbelos cut off by the line perpendicular to the base through the common point of the two small semicircles are equal. The circles have been known as Archimedes' Twin Circles. More than 2200 years after Archimedes, L. Bankoff (1974) has found another circle equal to the twins. In 1999 a large number of additional circles of the same radius has been reported by Dodge et al. More recently, F. Power described another quadruplet of circles that should be adopted into the family. The construction and the proof are exceedingly simple.

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Form the semicircles on diameters $AB, AC, BC,\,$ as in the applet above. Let the two smaller semicircles have radii $r_{1}\,$ and $r_{2},\,$ so that the radius r of the big semicircle satisfies

$r = r_{1} + r_{2}.$

Recollect that, according to Proposition 5 of the Book of Lemmas the common radius of Archimedes' Twins equals

$\displaystyle \frac{r_{1}\cdot r_{2}}{r} = \frac{r_{1}\cdot r_{2}}{r_{1} + r_{2}}.$

F. Power's construction is as follows. Let $E\,$ be the center and $D\,$ the midpoint of the semicircle of radius $r_{1}.\,$ (Naturally, a similar construction works for the other semicircle.) Let $O\,$ be the center of the big semicircle. Then, by the Pythagorean theorem,

$OD^{2} = r_{1}^{2} + r_{2}^{2}.$

There are two equal circles that touch the big semicircle and $OD\,$ at $D.\,$ Let $L\,$ be the center and $x\,$ the radius of one of them. Let $K\,$ denote the point of tangency of the latter with the big semicircle. Then applying the Pythagorean theorem a second time,

$OL^{2} = x^{2} + OD^{2} = x^{2} + r_{1}^{2} + r_{2}^{2}.$

On the other hand,

$OL = OK - x = r - x = r_{1} + r_{2} - x.$

Combining the two gives

$(r_{1} + r_{2} - x)^{2} = x^{2} + r_{1}^{2} + r_{2}^{2},$

from which

$- 2x(r_{1} + r_{2}) + 2r_{1}r_{2} = 0,$

$x = r_{1}\cdot r_{2}/(r_{1} + r_{2}),$

which is exactly the radius of the Archimedes' Twins.

References

L. Bankoff, Are the Twin Circles of Archimedes Really Twin, Mathematics Magazine, Vol. 47, No. 4 (Sept., 1974), 214-218
C.W. Dodge, T. Schoch, P.Y. Woo, and P. Yiu, Those ubiquitous Archimedean circles, Mathematics Magazine, Vol. 72 (1999), 202-213.
F. Power, Some More Archimedean Circles in the Arbelos, Forum Geometricorum, Vol. 5 (2005) 133-134.

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