In[]:=
ClearAll["`*"]​​intersectionColors={Blue,Green};​​circleColors={Blue,Green,Black};​​line=Graphics[{Black,Line[{{-2,2},{2,-2}}]}];​​makeGraphics[x_]:=Module[{diskList,intersections,cirlces,allArgs},​​diskList=Disk[{#,-#},1]&/@{-1,1,x};​​​​intersections=​​MapThread[Region@Style[RegionIntersection[#1,Last@diskList],#2]&,{Most@diskList,intersectionColors}];​​​​cirlces=MapThread[Region@Style[#1,#2]&,{Circle@@@diskList,circleColors}];​​allArgs=Flatten[{intersections,cirlces,{line}}];​​​​Show@@allArgs​​]​​Manipulate[Labeled[makeGraphics[x],"center of middle circle"<>ToString@{x,-x}],{x,-1,1}]​​
Out[]=
​
x
makeGraphics[-0.21]
center of middle circle{-0.21, 0.21}
​
In[]:=
(*haventfiguredouthowtogetSolveorReducetogiveanexactvalue,buthereisanumericalapproximationtofindingwheretheblueareais4timesthegreenarea*)​​eq=Simplify
Area@RegionIntersection[Disk[{-1,1},1],Disk[{x,-x},1]]
Area@RegionIntersection[Disk[{1,-1},1],Disk[{x,-x},1]]
==4,-1<x<1​​xSoln=x/.FindRoot[eq,{x,0}]​​EuclideanDistance@@eq/.x->xSoln
Out[]=
-(1+x)
1-2x-
2
x
+2ArcCos
1+x
2

(-1+x)
1+2x-
2
x
+2ArcCos-
-1+x
2

4
Out[]=
-0.183674
Out[]=
2.84217×
-14
10