ThirdJapaneseTheorem::usage="​
​​​Threecirclesaretangenttoastraightlineandbetweenthemselves.Itisfulfilledthat:​
1
r
3
=
1
r
1
+
1
r
2
Which is obtained by successively applying the Pythagoras theorem:​
2
x
2
+ ​
r
1
-
r
2
2
)
= ​
r
1
+
r
2
2
)
​, giving:
x
2
= 2
r
1
r
2
​, then:​
2
x
3
+ ​
r
1
-
r
3
2
)
= ​
r
1
+
r
3
2
)
and ​
x
3
-
x
2
2
)
+ ​
r
2
-
r
3
2
)
= ​
r
2
+
r
3
2
)
​, giving: ​
x
3
=
2
r
1
r
2
r
1
+
r
2
and
r
3
=
r
1
r
2
2

r
1
+
r
2

. The latter can be rewritten as:
1
r
3
=
1
r
1
+
1
r
2
.References: Weisstein, Eric W. Tangent Circles from MathWorld; Personal.;
In[]:=
TangentCircles[r1_,r2_]:=​​Module{},Ifr1<0∨r2<0∨r1==0∨r2==0,Print[PasteButton["The radii have to be different to zero and positives"]],​​Manipulatex2=2
r1r2
;​​x3=
2r1
r2
r1
+
r2
;r3=
r1r2
2

r1
+
r2

;​​Show[{ParametricPlot[{r1Cos[θ],r1Sin[θ]+r1},{θ,0,k1},PlotStyle{Black,Thick}],ParametricPlot[{r2Cos[θ]+x2,r2Sin[θ]+r2},{θ,0,k1},PlotStyle{Black,Thick}],ParametricPlot[{r3Cos[θ]+x3,r3Sin[θ]+r3},{θ,0,k1},PlotStyle{Black,Thick}],Graphics[{Black,Thick,InfiniteLine[{{0,0},{x2,0}}]}]},AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize900,PlotRangeAll],Style["Third Japanese Theorem:",Bold,Medium],{{k1,0.00002,"Value (k1)"},0.00001,2π,0.00001},ControlPlacementTop​​​​
In[]:=
TangentCircles[4,5]
Out[]=
​
Third Japanese Theorem:
Value (k1)