Archimedes' quadruplets

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. [ citation
last=Power
first=Frank
title=Forum Geometricorum
volume=5
chapter=Some More Archimedean Circles in the Arbelos
date=2005
publication-date=2005-11-02
editor-last=Yiu
editor-first=Paul
pages=133-134
isbn=1534-1178
url=http://forumgeom.fau.edu/FG2005volume5/FG200517.ps
accessdate=2008-04-13]

Construction

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".

Proof of congruency

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

:$frac\{r\_1cdot\; r\_2\}\{r\}.$

By the Pythagorean theorem:

:$left(HE\; ight)^2=left(r\_1\; ight)^2+left(r\_2\; ight)^2.$

Then, create two circles with centers "J_i" perpendicular to "HE", tangent to the large semicircle at point "L_i", tangent to point "E", and with equal radii "x". Using the Pythagorean theorem:

:$left(HJ\_i\; ight)^2=left(HE\; ight)^2+x^2=left(r\_1\; ight)^2+left(r\_2\; ight)^2+x^2$

Also:

:$HJ\_i=HL\_i-x=r-x=r\_1+r\_2-x~$

Combinding these gives:

:$left(r\_1\; ight)^2+left(r\_2\; ight)^2+x^2=left(r\_1+r\_2-x\; ight)^2$

Expanding, collecting to one side, and factoring:

:$2r\_1r\_2-2xleft(r\_1+r\_2\; ight)=0$

Solving for "x":

:$x=frac\{r\_1cdot\; r\_2\}\{r\_1+r\_2\}=frac\{r\_1cdot\; r\_2\}\{r\}$

Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas. [citeweb
last=Bogomolny
first=Alexander
title=Archimedes' Quadruplets
url=http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesQuadruplets.shtml
accessdate=2008-04-13]

References