Arbelos

Author

Eric W. Weisstein
June 5, 2015

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Laminae/Arbelos.nb.

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

©2015 Wolfram Research, Inc. except for portions noted otherwise

Code

Arbelos[r_:0.6,o_:{0,0},s_:1]:={{GrayLevel[0.6],Disk[o+s{1/2,0},s/2,{0,Pi}]},Disk[o+s{r/2,0},s(r/2),{0,Pi}],Disk[o+s{(1+r)/2,0},s((1-r)/2),{0,Pi}],{GrayLevel[1],Disk[o+s{r/2,0},s(r/2),{0,Pi}],Disk[o+s{(1+r)/2,0},s((1-r)/2),{0,Pi}]},Circle[o+s{r/2,0},s(r/2),{0,Pi}],Circle[o+s{(1+r)/2,0},s((1-r)/2),{0,Pi}],Circle[o+s{1/2,0},s/2,{0,Pi}],Line[{o,o}+s{{0,0},{1,0}}]}

ArbelosCircleRadius[r_]:=r((1-r)/2)

ArbelosSimilarityPoint[r_]:=Point[{(r+3r^2-2r^3)/2,(1-r)(r/2)}]

ArbelosCircle[0,r_]:=Circle[{1/2,0},1/2]
ArbelosCircle[1,r_]:=Circle[{r/2,0},r/2]
ArbelosCircle[2,r_]:=Circle[{(1+r)/2,0},(1-r)/2]

ArbelosSemicircle[0,r_]:=Circle[{1/2,0},1/2,{0,Pi}]
ArbelosSemicircle[1,r_]:=Circle[{r/2,0},r/2,{0,Pi}]
ArbelosSemicircle[2,r_]:=Circle[{(1+r)/2,0},(1-r)/2,{0,Pi}]

ArbelosCircle[1,1,r_]:=Circle[{r((1+r)/2),r Sqrt[1-r]},r((1-r)/2)]
ArbelosCircle[1,2,r_]:=Circle[{r((3-r)/2),Sqrt[r](1-r)},r((1-r)/2)]
ArbelosCircle[2,1,r_]:=Circle[{r^2+r((1-r)/2),r Sqrt[1-r^2]},r((1-r)/2)]
ArbelosCircle[2,2,r_]:=Circle[{r(2-r)-r((1-r)/2),(1-r)Sqrt[r(2-r)]},r((1-r)/2)]
ArbelosCircle[3,1,r_]:=Circle[{r((1+3r-2r^2)/2),r(1-r)Sqrt[(2-r)(1+r)]},r((1-r)/2)]
ArbelosCircle[3,2,r_]:=Circle[{r,r((1-r)/2)},r((1-r)/2)]
ArbelosCircle[3,3,r_]:=Circle[{r((1+3r-2r^2)/2),r(1-r)Sqrt[1+r-r^2]},r((1-r)/2)]
ArbelosCircle[4,1,r_]:=Circle[{(r(1+3r-5r^3+2r^4))/(1+4r-4r^2),((-1-2r+2r^2)Sqrt[r(1-r)(2r^2-3r-1)(2r^2-r-2)])/(2(-1-4r+4r^2))},r((1-r)/2)]
ArbelosCircle[4,2,r_]:=Circle[{(-(1/4))r(-3-3r+2r^2),r((1-r)/4)Sqrt[(1+2r)(3-2r)]},r((1-r)/2)]
ArbelosCircle[4,3,r_]:=Circle[{(1/4)(1+(2r-1)Sqrt[1+4r-4r^2])+r/2,r((1-r)/2)},r((1-r)/2)]
ArbelosCircle[5,1,r_]:=Circle[{r((1+r)/2),0},r((1-r)/2)]
ArbelosCircle[5,2,r_]:=Circle[{r((3-r)/2),0},r((1-r)/2)]
ArbelosCircle[5,3,r_]:=Circle[{r((1+3r-2r^2)/2),r((1-r)/2)},r((1-r)/2)]
ArbelosCircle[6,1,r_]:=Circle[{1/2,(1-r+r^2)/2},r((1-r)/2)]
ArbelosCircle[6,2,r_]:=Circle[{(r(1-r+2r^2))/(2(1-2r+2r^2)),-(((-1+r)r(1-r+r^2))/(1-2r+2r^2))},r((1-r)/2)]
ArbelosCircle[6,3,r_]:=Circle[{r,r((1-r)/2)},r((1-r)/2)]

ArchimedeanCircles[r_]:={Circle[{r(1+r)/2,r Sqrt[1-r]},r(1-r)/2],Circle[{r(3-r)/2,(1-r)Sqrt[r]},r(1-r)/2]}

PappusChain[n_,r_]:=PappusCircle[#,r]&/@Range[n]

PappusCircle[n_,r_]:=Circle[PappusCtr[n,r],((1-r)r)/(2(n^2(1-r)^2+r))]
PappusCircle[0,r]:=ArbelosCircle[2,r]

PappusCtr[n_,r_]:={(r(1+r))/(2(n^2-(-1+2n^2)r+n^2r^2)),(n(1-r)r)/(n^2-(-1+2n^2)r+n^2r^2)}

PappusDisk[n_,r_]:={{GrayLevel[1],Disk@@PappusCircle[n,r]},PappusCircle[n,r]}

PappusEllipse[r_]:=Circle[{(1+r)/4,0},{(1+r)/4,Sqrt[r]/2},{0,Pi}]

PappusLetter[n_,r_,o_:{0,0}]:=Text[Subscript[P,n],PappusCtr[n,r]+o]

PappusTangentCircle[n_,r_]:=Module[{d=4(n^2-n+1)-(8n^2-8n+1)r+(2n-1)^2r^2},Circle[{r((7+r)/2),2(2n-1)r(1-r)}/d,r((1-r)/2/d)]]
PappusTangentCircle[1,n_,r_]:=Module[{d=12+3n(3n-4)(1-r)^2+r(4r-7)},Circle[{r((17+r)/2/d),3(3n-2)(1-r)(r/d)},r((1-r)/d/2)]]
PappusTangentCircle[2,n_,r_]:=Module[{d=9+3n(3n-2)(1-r)^2-r(1-r)},Circle[{r((17+r)/2/d),3(3n-1)(1-r)(r/d)},r((1-r)/d/2)]]
PappusTangentCircle[3,n_,r_]:=Module[{d=9-12n(-1+r)^2+12n^2(-1+r)^2+r(-1+4r)},Circle[{r((17+7r)/2/d),6(2n-1)(1-r)(r/d)},r((1-r)/d/2)]]

Definitions

ineq=(x^2)+(y^2)≤(a^2)/4&&y≥0&&(y^2)≥(a/2 (-1+2 r)-x) (a/2+x)&&(a^2)/4 (-1+2 r)+(x^2)+(y^2)≥a r x;

implreg=ImplicitRegion[ineq,{x,y}];

reg=RegionDifference[Disk[{0,0},a/2,{0,π}],RegionUnion[Disk[{a(r-1)/2,0},a r/2,{0,π}],Disk[{a r/2,0},a(1-r)/2,{0,π}]]];

reg2=RegionDifference[Disk[{0,0},a/2],RegionUnion[Disk[{a(r-1)/2,0},a r/2],Disk[{a r/2,0},a(1-r)/2]]];

assum=a>0&&0<r<1;

Plots

Graphics

V5

Show[Graphics[Arbelos[.6]],AspectRatio->Automatic,ImageSize->300]

V6

Graphics[Arbelos[.6]]

Entity Diagram

EntityValue[Entity["Lamina", "Arbelos"], "Diagram"]

Entity Image

EntityValue[Entity["Lamina","Arbelos"],"Image"]

Entity ImplicitRegion

Show[DiscretizeRegion[EntityValue[Entity["Lamina","Arbelos"],"ImplicitRegion"][1,.2]],ImageSize->Small]

Entity ParametricRegion

EntityValue[Entity["Lamina","Arbelos"],"ParametricRegion"]

Entity Region

Show[DiscretizeRegion[EntityValue[Entity["Lamina","Arbelos"],"Region"][1,.2]],ImageSize->Small]

DiscretizeRegion

With[{a=1,r=.3},DiscretizeRegion[reg,Method->"Boolean"]]

With[{a=1,r=.3},DiscretizeRegion[reg,Method->"Boolean"]]

RegionPlot

x^2+y^2≤a^2&&
y≥0&&
(x-(a r-a))^2+y^2≥(a r)^2&&
(x-(a-a(1-r)))^2+y^2≥(a(1-r))^2

FullSimplify[%,a>0&&0<r<1]

Block[{a=1,r=.7},RegionPlot[ineq,{x,-a/2,a/2},{y,0,a/2},AspectRatio->Automatic,MaxRecursion->6,PlotPoints->50]]

Properties

Area

Entity

EntityValue[Entity["Lamina", "Arbelos"], "Area"][a,r]

FullSimplify[Subtract@@{(1/4) (a^2) π (1-r) r,-(1/4) (a^2) π (-1+r) r}]

ByteCount/@{(1/4) (a^2) π (1-r) r,-(1/4) (a^2) π (-1+r) r}

Simplify`SimplifyCount/@{(1/4) (a^2) π (1-r) r,-(1/4) (a^2) π (-1+r) r}

Area

Assuming[assum,Area[EntityValue[Entity["Lamina","Arbelos"],"ImplicitRegion"][a,r]]//FullSimplify]//Timing

Assuming[assum,Area[EntityValue[Entity["Lamina","Arbelos"],"Region"][a,r]]//FullSimplify]//Timing

V7 [GenerateConditions -> True]

(area=FullSimplify[Integrate[Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True,Assumptions->a>0&&0<r<1],0<r<1&&a>0])//Timing

FullSimplify[Last[%3],a>0&&0<r<1]

...pretty sure that's wrong...

V8 [GenerateConditions -> Automatic]

$Version

(area=Assuming[a>0&&0<r<1,FullSimplify[Area[ineq[x,y],{x,y}]]])//Timing

(-(1/8) (a^2) (2 π (-1+r) r+ArcSin[1-2 r]+ArcTan[((-1+2 r)/(2 (Sqrt[-(-1+r) r])))])/.r->#)&/@Range[.1,.9,.1]/a^2

Assuming[a>0&&0<r<1,FullSimplify[Integrate[Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]]]//Timing

area=-(1/4) (a^2) π (-1+r) r;

V8 [GenerateConditions -> True]

$Version

ineq=x^2+y^2≤a^2&&y≥0&&y^2≥(a (-1+2 r)-x) (a+x)&&a^2 (-1+2 r)+x^2+y^2≥2 a r x;

(area=Assuming[a>0&&0<r<1,Integrate[Boole[ineq],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True]])//Timing

{area/.r->1/8.,area/.r->7/8.}/.a->1

area2=FullSimplify[area,a>0&&0<r<1]

{area2/.r->1/8.,area/.r->7/8.}/.a->1

...first part correct; 2nd part wrong...

RegionMeasure V10.0.0

RegionMeasure[reg]

Assuming[assum,reg2]/2

RegionMeasure V10.0.1

Assuming[assum,FullSimplify@RegionMeasure[reg]]

Assuming[assum,FullSimplify@RegionMeasure[reg2]]/2

Assuming[assum,FullSimplify@RegionMeasure[implreg]]

Perimeter

Factor[Pi a+Pi r/2 a+Pi (1-r)/2 a]

Bounding Box

ineq=x^2+y^2≤a^2/4&&y≥0&&y^2≥(a/2 (-1+2 r)-x) (a/2+x)&&a^2/4 (-1+2 r)+x^2+y^2≥a r x;

Assuming[a>0&&0<r<1,FullSimplify[With[{vars={x,y}},Through[{MinValue,MaxValue}[{#,ineq},vars]]&/@vars]]]

Centroid

V5.0

(cog=FullSimplify[(1/(π (a^2)r(1-r)))Integrate[{x,y}Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True,Assumptions->a>0&&0<r<1],0<r<1&&a>0])//Timing

V5.2

(cog=FullSimplify[(1/(π (a^2)r(1-r)))Integrate[{x,y}Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True,Assumptions->a>0&&0<r<1],0<r<1&&a>0])//Timing

V6

(cog=FullSimplify[(1/(π (a^2)r(1-r)))Integrate[{x,y}Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True,Assumptions->a>0&&0<r<1],0<r<1&&a>0])//Timing

...wrong...

V7

(cog=FullSimplify[(1/(π (a^2)r(1-r)))Integrate[{x,y}Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True,Assumptions->a>0&&0<r<1],0<r<1&&a>0])//Timing

...wrong...

V8

Assuming[a>0&&0<r<1,FullSimplify[Centroid[ineq[x,y],{x,y}]]]//Timing

(cog=FullSimplify[(1/area)Integrate[{x,y}Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True,Assumptions->a>0&&0<r<1],0<r<1&&a>0])//Timing

Block[{a=1,r=.2},(1/area)NIntegrate[{x,y}Boole[ineq[x,y]],{x,-a/2,a/2},{y,0,a/2}]]

{(1/4) a (-1+2 r),(a/π)}/.{a->1.,r->.2}

GeneralizedDiameter

$Version

ineq[x_,y_]:=x^2+y^2≤a^2&&
y≥0&&
(x-(a r-a))^2+y^2≥(a r)^2&&
(x-(a-a(1-r)))^2+y^2≥(a(1-r))^2

ineqs=FullSimplify[And@@ineq@@@{{x1,y1},{x2,y2}},a>0&&0<r<1]/.a->1

Maximize[{(x1-x2)^2+(y1-y2)^2,ineqs&&0<r<1},{x1,x2,y1,y2},Reals]//Timing

...ran overnight...

InertiaTensor

V7

Integrate[AreaInertiaTensor[{x,y}] Boole[ineq[x,y]],{x,-Infinity,Infinity},{y,-Infinity,Infinity},GenerateConditions->True,Assumptions->a>0&&0<r<1]//Timing

FullSimplify[Last[%],a>0&&0<r<1]

Refine[%,r>0]

...pretty sure that's wrong...

V8

Assuming[a>0&&0<r<1,FullSimplify[AreaInertiaTensor[ineq[x,y],{x,y}]]]//Timing

(-(1/64) (a^4) π (-1+r) r (2+5 (-1+r) r)/.r->#)&/@{.1,.2,.8,.9}//Chop

(-(1/256) (a^4) (π (1+4 (-1+r) r (2+5 (-1+r) r))+2 I ArcCosh[1-2 r]+2 ArcTan[((-1+2 r)/(2 (Sqrt[-(-1+r) r])))])/.r->#)&/@{.1,.2,.8,.9}//Chop

FullSimplify[Subtract@@{-(1/64) (a^4) π (-1+r) r (2+5 (-1+r) r),-(1/256) (a^4) (π (1+4 (-1+r) r (2+5 (-1+r) r))+2 I ArcCosh[1-2 r]+2 ArcTan[((-1+2 r)/(2 (Sqrt[-(-1+r) r])))])},0<r<1]

Plot[Through[{Re,Im}[π+2 I ArcCosh[1-2 r]+2 ArcTan[((-1+2 r)/(2 (Sqrt[-(-1+r) r])))]]],{r,0,1},Evaluated->True]

FullSimplify[Subtract@@{-(1/256) (a^4) (π (1+4 (-1+r) r (2+5 (-1+r) r))+2 I ArcCosh[1-2 r]+2 ArcTan[((-1+2 r)/(2 (Sqrt[-(-1+r) r])))]),-(1/256) (a^4) (π+2 I ArcCosh[1-2 r]+2 ArcTan[((-1+2 r)/(2 (Sqrt[-(-1+r) r])))])}]

i={{-(1/64) (a^4) π (-1+r) r (2+(-1+r) r),(1/24) (a^4) (-1+r) r (-1+2 r)},{(1/24) (a^4) (-1+r) r (-1+2 r),-(1/64) (a^4) π (-1+r) r (2+5 (-1+r) r)}};

i//TraditionalForm

RadiiOfGyration

FullSimplify[(Sqrt[((Tr[i,List])/area)]),a>0&&0<r<1]

Regions

EntityValue[Entity["Lamina","Arbelos"],"ImplicitRegion"][a,r]

EntityValue[Entity["Lamina","Arbelos"],"ParametricRegion"]

EntityValue[Entity["Lamina","Arbelos"],"Region"][a,r]

Vertical Line Intersects at Right Vertex

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"}
}
},
Show[Graphics[{
Arbelos[r],
Line[{{r,0},{r,Sqrt[r(1-r)]}}],
PerpSquare[Line/@Map[Coordinates,{{a,b},{b,d}},{2}],.03,3],
PerpSquare[Line/@Map[Coordinates,{{a,b},{b,d}},{2}],.03,4],
PerpSquare[Line/@Map[Coordinates,{{a,d},{d,c}},{2}],.03,4],
{Dashing[{.02}],Line[Coordinates/@{a,d,c}]},
PointLabels[pts,LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]
]

(Subscript[C, 1]) (Archimedean Circles}

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],

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]
]

(Subscript[C, 2])

Block[{r=.6,
a=Point[{0,0}],
ap=Point[{r^2,r Sqrt[1-r^2]}],
b=Point[{r,0}],
cp=Point[{r(2-r),(1-r)Sqrt[r(2-r)]}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
c1=Point[{r^2+r(1-r)/2,r Sqrt[1-r^2]}],
c2=Point[{r(2-r)-r(1-r)/2,(1-r)Sqrt[r(2-r)]}],
pts={
{a,"A",LabelOffset->{0,-.04}},
{b,"B",LabelOffset->{0,-.04}},
{c,"C",LabelOffset->{0,-.04}},
{d,"D"},
{ap,"A'",LabelOffset->{-.03,.02}},
{cp,"C'"},
{c1,"(Subscript[C, 2])"},
{c2,"(Subscript[C, 2])'"}
}
},
Show[Graphics[{
Arbelos[r],
Line[{{r,0},{r,Sqrt[r(1-r)]}}],
{{GrayLevel[.8],Disk@@@#},#}&[{Circle[Coordinates[c1],r(1-r)/2],Circle[Coordinates[c2],r(1-r)/2]}],
PointLabels[pts,LabelOffset->.03,Background->None],
{Dashing[{.02}],
Circle[{0,0},r,{0,ArcTan[Sqrt[1-r^2]/r]}],
Circle[{1,0},1-r,{Pi-ArcTan[(1-r)Sqrt[r(2-r)]/(1-(r(2-r)))],Pi}]
}
}],AspectRatio->Automatic,PlotRange->All,ImageSize->300]
]

FullSimplify[{{r^2+r(1-r)/2,r Sqrt[1-r^2]},{r(2-r)-r(1-r)/2,(1-r)Sqrt[r(2-r)]}}]

(Subscript[C, 3])

Union[Flatten[FullSimplify[Table[Solve[Simplify[{
(x-x1)^2+(y-y1)^2-(R+(-1)^i  r1)^2==0,
(x-x2)^2+(y-y2)^2-(R+(-1)^j  r2)^2==0,
(x-x3)^2+(y-y3)^2-(R+(-1)^k  r3)^2==0} /. 
{x1->0,y1->0,r1->r,
x2->1,y2->0,r2->1-r,
x3->1/2,y3->0,r3->1/2}
]//Flatten,{x,y,R}],
{i,2},{j,2},{k,2}]],3]]

Block[{r=.6,
a=Point[{0,0}],
ap=Point[{r^2,r Sqrt[1-r^2]}],
b=Point[{r,0}],
cp=Point[{r(2-r),(1-r)Sqrt[r(2-r)]}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
c1=CircleCenter[ArbelosCircle[3,1,r]],
c2=CircleCenter[ArbelosCircle[3,2,r]],
pts={
{a,"A",LabelOffset->{0,-.04}},{b,"B",LabelOffset->{0,-.04}},{c,"C",LabelOffset->{0,-.04}},{d,"D"},{ap,"A'"},{cp,"C'"},{c1,"(Subscript[C, 3])"},
	{c2,"C(Subscript[', 3])",LabelOffset->{-.04,-.01}}
}
},
Show[Graphics[{
Arbelos[r],
{{GrayLevel[.8],Disk@@@#},#}&[ArbelosCircle[3,#,r]&/@Range[2]],
{Dashing[{.02}],
Circle[{0,0},r,{0,ArcTan[Sqrt[1-r^2]/r]}],
Circle[{1,0},1-r,{Pi-ArcTan[(1-r)Sqrt[r(2-r)]/(1-(r(2-r)))],Pi}]
},
Line[{{r,0},{r,Sqrt[r(1-r)]}}],
PointLabels[pts,LabelOffset->.03,Background->None]
}],AspectRatio->Automatic,PlotRange->All,ImageSize->300]
]

Block[{r=.6,
a=Point[{0,0}],
ap=Point[{r^2,r Sqrt[1-r^2]}],
b=Point[{r,0}],
cp=Point[{r(2-r),(1-r)Sqrt[r(2-r)]}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
c1=CircleCenter[ArbelosCircle[3,1,r]],
c3=CircleCenter[ArbelosCircle[3,3,r]],
bp=Point[{(1/2) r (1+3 r-2 (r^2)),0}],
dp=Point[{(1/2) r (1+3 r-2 (r^2)),(Sqrt[(1/4)-(1/4) (((-1+r+3 (r^2)-2 (r^3)))^2)])}],
pts={
{a,"A",LabelOffset->{0,-.04}},
{b,"B",LabelOffset->{-.01,-.04}},
{c,"C",LabelOffset->{0,-.04}},
{d,"D",LabelOffset->{0,.04}},
{ap,"A'",LabelOffset->{-.02,.03}},
{bp,"B'",LabelOffset->{0.01,-.04}},
{cp,"C'"},
{dp,"D'",LabelOffset->{.02,.04}},
{c3,"C'(Subscript[', 3])"},
{c1,"(Subscript[C, 3])"}}
},
Show[Graphics[{
Arbelos[r],
{GrayLevel[.8],Disk@@@(ArbelosCircle[3,#,r]&/@{1,3})},ArbelosCircle[3,3,r],
{Dashing[{.02}],
ArbelosCircle[3,1,r],
Circle[{0,0},r,{0,ArcTan[Sqrt[1-r^2]/r]}],
Circle[{1,0},1-r,{Pi-ArcTan[(1-r)Sqrt[r(2-r)]/(1-(r(2-r)))],Pi}]
},
Line[{{r,0},{r,Sqrt[r(1-r)]}}],
PointLabels[pts,LabelOffset->.03,Background->None],
Line[Coordinates/@{bp,dp}]
}
],AspectRatio->Automatic,PlotRange->All,ImageSize->300]
]

(Subscript[C, 4])

CircleCenter[ArbelosCircle[3,3,r]][[1]]

Block[{r=.6,
a=Point[{0,0}],
ap=Point[{r^2,r Sqrt[1-r^2]}],
b=Point[{r,0}],
cp=Point[{r(2-r),(1-r)Sqrt[r(2-r)]}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
p=Point[{r/2,0}],
q=Point[{r/2+1/2,0}],
m=Point[{r/2+1/4,0}],
c1=CircleCenter[ArbelosCircle[4,1,r]],
c2=CircleCenter[ArbelosCircle[4,2,r]],
c3=CircleCenter[ArbelosCircle[3,3,r]],
bp=Point[{(1/2) r (1+3 r-2 (r^2)),0}],
dp=Point[{(1/2) r (1+3 r-2 (r^2)),(Sqrt[(1/4)-(1/4) (((-1+r+3 (r^2)-2 (r^3)))^2)])}],
pts={
{a,"A",LabelOffset->{0,-.04}},
{b,"B",LabelOffset->{-.01,-.04}},
{c,"C"},
{Point[{r(1+3r-2r^2)/2,r(1-r)/2Sqrt[(1+2r)(3-2r)]}],"U"},
{p,"P",LabelOffset->{0,-.04}},
{q,"Q",LabelOffset->{0,-.04}},
{m,"M",LabelOffset->{0,-.04}},
{bp,"B'",LabelOffset->{.01,-.04}},
{dp,"D'"},
{c1,"(Subscript[C, 4])"},
{c2,"C(Subscript[', 4])"}
}
},
Show[Graphics[{
Arbelos[r],
{{GrayLevel[.8],Disk@@@#},#}&[ArbelosCircle[4,#,r]&/@Range[2]],Line[Coordinates/@{m,dp}],
PointLabels[pts,LabelOffset->.03,Background->None],
Line[Coordinates/@{bp,dp}],
{Dashing[{.02}],
{ArbelosCircle[4,1,r]},
Circle[{1/4+r/2,0},1/4,{0,Pi}]
}
}],AspectRatio->Automatic,PlotRange->All,ImageSize->300]
]

CircleCenter[ArbelosCircle[3,3,r]]

Block[{r=.6,
a=Point[{0,0}],
ap=Point[{r^2,r Sqrt[1-r^2]}],
b=Point[{r,0}],
cp=Point[{r(2-r),(1-r)Sqrt[r(2-r)]}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
p=Point[{r/2,0}],
q=Point[{r/2+1/2,0}],
m=Point[{r/2+1/4,0}],
c1=CircleCenter[ArbelosCircle[4,1,r]],
c2=CircleCenter[ArbelosCircle[4,2,r]],
c3=CircleCenter[ArbelosCircle[3,3,r]],
x=Point[{1/4(1+(2r-1)Sqrt[1+4r-4r^2])+r/2,r(1-r)}],
bp=Point[{(1/2) r (1+3 r-2 (r^2)),0}],
c4=CircleCenter[ArbelosCircle[4,3,r]],
pts={
{a,"A",LabelOffset->{0,-.04}},
{b,"B",LabelOffset->{0,-.04}},
{c,"C",LabelOffset->{0,-.04}},
{p,"P",LabelOffset->{0,-.04}},
{q,"Q",LabelOffset->{0,-.04}},
{m,"M",LabelOffset->{0,-.04}},
{x,"X"},
{c4,"C'(Subscript[', 4])"}
	}
},
Show[Graphics[{
Arbelos[r],
{Dashing[{.02}],
{{GrayLevel[.8],Disk@@#},#}&[Circle[{1/4(1+(2r-1)Sqrt[1+4r-4r^2])+r/2,r(1-r)},(Sqrt[1+4r-4r^2]-1)/4]],
Circle[{1/4+r/2,0},1/4,{0,Pi}],
PointLabels[pts,LabelOffset->.03,Background->None],
ArbelosCircle[4,3,r],
PerpSquare[Line/@Map[Coordinates,{{p,x},{q,x}},{2}],.03,1],
Line[Coordinates/@{p,x,q}]
}
}],AspectRatio->Automatic,PlotRange->All,ImageSize->300]
]

(Subscript[C, 5])

ArbelosSimilarityPoint[r]

Block[{r=.6,
a=Point[{0,0}],
b=Point[{r,0}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
c3=CircleCenter[ArbelosCircle[3,3,r]],
bp=Point[{(1/2) r (1+3 r-2 (r^2)),0}],dp=Point[{(1/2) r (1+3 r-2 (r^2)),(Sqrt[(1/4)-(1/4) (((-1+r+3 (r^2)-2 (r^3)))^2)])}],
pts1={
{a,"A"},
{b,"B"},
{c,"C"},
{bp,"B'"},
{dp,"D'"},
{CircleCenter[ArbelosCircle[5,1,r]],"(Subscript[C, 5])"},{CircleCenter[ArbelosCircle[5,2,r]],"C(Subscript[', 5])"}
},
e=Point[{r/2,r/2}],
f=Point[{r(1+r)/2,r(1-r)/2}],
g=Point[{r^2,0}],
h=Point[{r^2/2,r^2/2}],
ep=Point[{r(2-r),0}],
fp=Point[{r(3-r)/2,r(1-r)/2}],
gp=Point[{(1+r)/2,(1-r)/2}],
hp=Point[{(1+2r-r^2)/2,(1-r)^2/2}],
pts2={
{a,"A"},
{b,"B"},
{c,"C"},
{bp,"B'"},
{dp,"D'"},
{ArbelosSimilarityPoint[r],"C'(Subscript[', 5])"},
{e,"E"},
{f,"F"},
{g,"G"},
{h,"H"},
{ep,"E'"},
{fp,"F'"},
{gp,"G'"},
{hp,"H'"}
}
},
Show[GraphicsArray[
{Graphics[{
Arbelos[r],
Arbelos[r,{0,0},r],
Arbelos[r,{r,0},1-r],
Line[Coordinates/@{bp,dp}],
{ArbelosCircle[5,1,r],ArbelosCircle[5,2,r]},
PointLabels[pts1,LabelOffset->.03]
},AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.2,.6}}],
Graphics[{
Arbelos[r],
Arbelos[r,{0,0},r],
Arbelos[r,{r,0},1-r],
Line[Coordinates/@{bp,dp}],
PointLabels[pts2,LabelOffset->.03,Background->None],
{Thickness[.01],ClosedLine[Coordinates/@{e,f,g,h}],ClosedLine[Coordinates/@{ep,fp,gp,hp}]},
{Dashing[{.02}],{ArbelosCircle[5,3,r]}},
{Dashing[{.015}],Line[Coordinates/@{e,ep}],Line[Coordinates/@{f,fp}],Line[Coordinates/@{g,gp}],Line[Coordinates/@{h,hp}]}
},
AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.2,.6}}
]}],ImageSize->500,GraphicsSpacing->-.01]
]

(Subscript[C, 6])

Block[{r=.6,
a=Point[{0,0}],
b=Point[{r,0}],
c=Point[{1,0}],
d=Point[{r,Sqrt[r(1-r)]}],
m=Point[{1/2,0}],
mp=Point[{1/2,1/2}],
e=Point[{r/2,r/2}],
gp=Point[{(1+r)/2,(1-r)/2}],
i=Point[{1/2,1/2-r+r^2}],
j=Point[{r,r(1-r)}],
l=Point[{((r^2)/(1-2 r+2 (r^2))),-(((-1+r) r)/(1-2 r+2 (r^2)))}],
pts1={
{a,"A"},
{b,"B"},
{c,"C"},
{d,"D"},
{e,"E"},
{gp,"G'"},
{m,"M"},
{mp,"M'"},
{i,"I"},
{j,"J"},
{l,"L"},
{CircleCenter[ArbelosCircle[6,1,r]],"(Subscript[C, 6])"},
{CircleCenter[ArbelosCircle[6,2,r]],"(Subscript[C, 6])'"}
},
pts2={
{a,"A"},
{b,"B"},
{c,"C"},
{d,"D"},
{e,"E"},
{gp,"G'"},
{m,"M"},
{mp,"M'"},
{i,"I"},
{j,"J"},
{CircleCenter[ArbelosCircle[6,3,r]],"(Subscript[C, 6])''"},
{Point[{r/(2-2r+r^2),r(1-r)/(2-2r+r^2)}],"(Subscript[P, A])"},
{Point[{r(1+r)/(1+r^2),r(1-r)/(1+r^2)}],"(Subscript[P, B])"},
{Point[{r^2/(1-2r+2r^2),r(1-r)/(1-2r+2r^2)}],"(Subscript[P, C])"}
}
},
Show[GraphicsArray[{
Graphics[{
Arbelos[r],
Line[Coordinates/@{b,d}],Line[Coordinates/@{m,mp}],
Line[Coordinates/@{e,gp}],
ArbelosCircle[6,1,r],ArbelosCircle[6,2,r],
PointLabels[pts1,Background->None,LabelOffset->.03],
{Dashing[{.02}],Line[Coordinates/@{m,l}],
Circle[{(1+2r)/4,1/4},Sqrt[2(1-2r+2r^2)]/4]}
},
AspectRatio->Automatic,PlotRange->All],
Graphics[{
Arbelos[r],PappusDisk[1,r],
Line[Coordinates/@{b,d}],Line[Coordinates/@{e,gp}],Line[Coordinates/@{m,mp}],
PointLabels[pts2,Background->None,LabelOffset->.03],
ArbelosCircle[6,3,r]
},AspectRatio->Automatic,PlotRange->All]
	}],ImageSize->450]
]

{x,y,z}/.Simplify[Solve[{
(x-r/2)^2+y^2==(r/2+R)^2,
(x-r-(1-r)/2)^2+y^2==((1-r)/2+R)^2,
(x-1/2)^2+y^2==(-1/2+R)^2},
{x,y,R}]]

{x,y}/.Solve[{
(x-1/2)^2+y^2==(1/2)^2,
(x-((r (1+r))/(2 (1-r+(r^2)))))^2+(y-r(1-r)/(1-r+r^2))^2==R^2} /.R->-(((-1+r) r)/(2 (1-r+(r^2)))) ,
{x,y}]