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In geometry, **Archimedes' quadruplets** are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles.^{[1]}^{[2]}^{[3]}

An arbelos is formed from three collinear points *A*, *B*, and *C*, by the three semicircles with diameters *AB*, *AC*, and *BC*. Let the two smaller circles have radii *r*_{1} and *r*_{2}, from which it follows that the larger semicircle has radius *r* = *r*_{1}+*r*_{2}. Let the points *D* and *E* be the center and midpoint, respectively, of the semicircle with the radius *r*_{1}. Let *H* be the midpoint of line *AC*. Then two of the four quadruplet circles are tangent to line *HE* at the point *E*, and are also tangent to the outer semicircle. The other two quadruplet circles are formed in a symmetric way from the semicircle with radius *r*_{2}.

According to Proposition 5 of Archimedes' *Book of Lemmas*, the common radius of Archimedes' twin circles is:

By the Pythagorean theorem:

Then, create two circles with centers *J _{i}* perpendicular to

Also:

Combining these gives:

Expanding, collecting to one side, and factoring:

Solving for *x*:

Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas.^{[4]}

**^**Power, Frank (2005), "Some More Archimedean Circles in the Arbelos", in Yiu, Paul (ed.),*Forum Geometricorum*, vol. 5 (published 2005-11-02), pp. 133–134, ISSN 1534-1178, retrieved 2008-04-13**^**Online catalogue of Archimedean circles**^**Clayton W. Dodge, Thomas Schoch, Peter Y. Woo, Paul Yiu (1999). "Those Ubiquitous Archimedean Circles". PDF.**^**Bogomolny, Alexander. "Archimedes' Quadruplets". Archived from the original on 12 May 2008. Retrieved 2008-04-13.

- Arbelos: Book of Lemmas, Pappus Chain, Archimedean Circle, Archimedes' Quadruplets, Archimedes' Twin Circles, Bankoff Circle, S. ISBN 1156885493