Archimedes' Circles

Author

Eric W. Weisstein
March 27, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/ArchimedesCircles.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ArchimedesCircles.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Derivation

(Subscript[C, 1])

Simplify[Solve[{
(Sqrt[(((x-r/2))^2)+(y^2)])==(r/2)+ρ,
(Sqrt[(((x-(1/2)))^2)+(y^2)])==(1/2)-ρ,
x+ρ==r},
{x,y,ρ}],0<r<1]

(Subscript[C, 2])

Simplify[Solve[{
(Sqrt[(((x-(r+(1-r)/2)))^2)+(y^2)])==((1-r)/2)+ρ,
(Sqrt[(((x-(1/2)))^2)+(y^2)])==(1/2)-ρ,
x-ρ==r},
{x,y,ρ}],0<r<1]

Figure

Figure for Distances

Block[{
r=.6,
a=Point[{0,0}],
b=Point[{r,0}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
c1=Point[{r(1+r)/2,r Sqrt[1-r]}],
c2=Point[{r(3-r)/2,(1-r)Sqrt[r]}],
pts={
{a,"A",LabelOffset->{0,-.04}},
{b,"B",LabelOffset->{0,-.04}},
{c,"C",LabelOffset->{0,-.04}},
{d,"D"},{c1,"(Subscript[C, 1])"},{c2,"C(Subscript[', 1])"},
{Midpoint[{a,b}],"(Subscript[M, AB])",LabelOffset->{0,-.04}},{Midpoint[{b,c}],"(Subscript[M, BC])",LabelOffset->{0,-.04}}
}
},
Show[Graphics[{
Arbelos[r],
Line[{{r,0},{r,Sqrt[r(1-r)]}}],
{GrayLevel[.8],Disk@@@ArchimedeanCircles[r]},
ArchimedeanCircles[r],
PerpSquare[{Line[Coordinates/@{a,b}],Line[{{c1[[1,1]],0},c1[[1]]}]},.03,4],
PerpSquare[{Line[Coordinates/@{b,c}],Line[{{c2[[1,1]],0},c2[[1]]}]},.03,3],
PointLabels[pts,LabelOffset->.03,Background->None],
{
Dashing[{.02}],
Line[{{c1[[1,1]],0},c1[[1]],(a[[1]]+b[[1]])/2}],Line[{{c2[[1,1]],0},c2[[1]],(b[[1]]+c[[1]])/2}]
}
}],AspectRatio->Automatic,PlotRange->All,ImageSize->300]
]

Figure

Block[{
r=.6,
a=Point[{0,0}],
b=Point[{r,0}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
c1=Point[{r(1+r)/2,r Sqrt[1-r]}],
c2=Point[{r(3-r)/2,(1-r)Sqrt[r]}],
pts={
{a,"A",LabelOffset->{0,-.04}},
{b,"B",LabelOffset->{0,-.04}},
{c,"C",LabelOffset->{0,-.04}},
{d,"D"},{c1,"(Subscript[C, 1])"},{c2,"(Subscript[C, 1])'"}
}
},
Show[Graphics[{
Arbelos[r],
Line[{{r,0},{r,Sqrt[r(1-r)]}}],
{GrayLevel[.8],Disk@@@ArchimedeanCircles[r]},
ArchimedeanCircles[r],
PointLabels[pts,LabelOffset->.03,Background->None]
}],AspectRatio->Automatic,PlotRange->All,ImageSize->300]
]

PerpSquare[Line/@Map[Coordinates,{{a,b},{b,d}},{2}],.03,3],