Soddy Circles

Author

Eric W. Weisstein
March 14, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Graphic

t=Triangle[{{0,0},{1,0},{.3,.8}}];
Show[Graphics[{
t,
{Green,TangentCircles[t]},
{Red,InnerSoddyCircle[t]},
{Blue,OuterSoddyCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{InnerSoddyCenter[t],"S",Red},
{OuterSoddyCenter[t],"S'",Blue}
}
],PointSize[.025],LabelOffset->.05]
},AspectRatio->Automatic]
]

Points at which Circles Touch

t=Triangle[{{0,0},{1,0},{.3,.6}}];
Show[Graphics[{
t,
{Blue,TangentCircles[t]},
(*
{Blue,OuterSoddyCircle[t],InnerSoddyCircle[t]},
*)
{Purple,Incircle[t]},
{Red,ContactTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ContactTriangle[t]],{"(Subscript[C, 1])","(Subscript[C, 2])","(Subscript[C, 3])"}}]
],PointSize[.022],LabelOffset->.055]
},AspectRatio->Automatic]
]

Radii

Solve[{aa+bb==c,aa+cc==b,bb+cc==a},{aa,bb,cc}]

Simplify[2Sum[(Subscript[κ, i])^2,{i,n+1}]-(Sum[(Subscript[κ, i]),{i,n+1}])^2 /. n->3]

FullSimplify[Solve[2Plus@@(#^2)==(Plus@@#)^2&[Array[ϵ,4]],ϵ[4]]]

(solns=FullSimplify[Solve[2Plus@@(#^2)==(Plus@@#)^2&[Array[ϵ,4]]/.ϵ[i_]:>1/r[i],r[4]],r[1]>0&&r[2]>0&&r[3]>0])//TraditionalForm

Factor[FullSimplify[Together//@(#[[1,2]]&/@(solns/.{r[1]->(-a+b+c)/2,r[2]->(a-b+c)/2,r[3]->(a+b-c)/2}))/.(Sqrt[-(((a-b-c) (a+b-c) (a-b+c))/(a+b+c))])->2r,TriangleQ[]]]

d1=-((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)+(a^2) r-2 a b r+(b^2) r-2 a c r-2 b c r+(c^2) r);

d1

Factor[Expand[d1-Coefficient[d1,r]r]]

Factor[Expand[8Inradius[]^2Semiperimeter[]]]

Factor[Coefficient[d1,r]]

8r^2 s+(-(a^2)+2 a b-(b^2)+2 a c+2 b c-(c^2))r

8r^2 s-r(a^2+b^2+c^2)+2r(a b+b c+c a)

Expand[d1-(8r^2 s-r(a^2+b^2+c^2)+2r(a b+b c+c a))/.{s->Semiperimeter[],r->Inradius[]}]

((a-b-c) (a+b-c) (a-b+c) r)/(2 ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)+(a^2) r-2 a b r+(b^2) r-2 a c r-2 b c r+(c^2) r))

(((a-b-c) (a+b-c) (a-b+c) r)/(-2(8r^2 s-r(a^2+b^2+c^2)+2r(a b+b c+c a))))

8Inradius[]^2Semiperimeter[]

Cancel[(((a-b-c) (a+b-c) (a-b+c) r)/(-2(8r^2 s-r(a^2+b^2+c^2)+2r(a b+b c+c a))))/.(a-b-c) (a+b-c) (a-b+c)->r^2 s]

((r^2 s)/(2(-8r s+(a^2+b^2+c^2)-2(a b+b c+c a))))

Together[Together[((4r^2 s)/(8r s-(a^2+b^2+c^2)+2(a b+b c+c a)))-((a-b-c) (a+b-c) (a-b+c) r)/(2 ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)+(a^2) r-2 a b r+(b^2) r-2 a c r-2 b c r+(c^2) r))]/.s->Semiperimeter[]/.r^3->Inradius[]^2 r/.r^2->Inradius[]^2]

d2

d2=-((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)-(a^2) r+2 a b r-(b^2) r+2 a c r+2 b c r-(c^2) r);

Factor[d2-Coefficient[d2,r]r]

Factor[Expand[8Inradius[]^2Semiperimeter[]]]

Factor[Coefficient[d2,r]]

Factor[Together[Together[((4r^2 s)/(-8r s-(a^2+b^2+c^2)+2(a b+b c+c a)))-(-(a-b-c) (a+b-c) (a-b+c) r)/(2 ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)-(a^2) r+2 a b r-(b^2) r+2 a c r+2 b c r-(c^2) r))]/.s->Semiperimeter[]/.r^3->Inradius[]^2 r/.r^2->Inradius[]^2]]

Factor[Together[Together[((4r^2 s)/(2(a b+b c+c a)-(a^2+b^2+c^2)-8r s))-(-(a-b-c) (a+b-c) (a-b+c) r)/(2 ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)-(a^2) r+2 a b r-(b^2) r+2 a c r+2 b c r-(c^2) r))]/.s->Semiperimeter[]/.r^3->Inradius[]^2 r/.r^2->Inradius[]^2]]

Both

{((4r^2 s)/(8r s+(2(a b+b c+c a)-(a^2+b^2+c^2)))),((4r^2 s)/(8r s-(2(a b+b c+c a)-(a^2+b^2+c^2))))}

FullSimplify[2(a b+b c+c a)-(a^2+b^2+c^2)>0,TriangleQ[]]

Groebner

eqns=Numerator/@Together//@{r^2-Inradius[]^2,R^2-Circumradius[]^2,s-Semiperimeter[]}

cos=Numerator/@Together//@{Cos[A]-DistanceTrilinears[Cos[A]],Cos[B]-DistanceTrilinears[Cos[B]],Cos[C]-DistanceTrilinears[Cos[C]]}

Factor/@GroebnerBasis[{2(a b+b c+c a)-(a^2+b^2+c^2),Sequence@@eqns},{a,b,c},MonomialOrder->EliminationOrder]

Factor[Together//@(Expand[(r-4 R) (r+4 R)]/.{r->Inradius[],R->Circumradius[]})]

Factor/@GroebnerBasis[{((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+10 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)),Sequence@@eqns},{a,b,c},MonomialOrder->EliminationOrder]

Circle Function

<<PlaneGeometry`

InnerSoddyCircle[t=Triangle[{{0,0},{1,0},{.3,.7}}]]

SoddyCircles[t=Triangle[{{0,0},{1,0},{.3,.7}}]]

{((4r^2 s)/(8r s+2(a b+b c+c a)-(a^2+b^2+c^2))),((4r^2 s)/(8r s-2(a b+b c+c a)+(a^2+b^2+c^2)))}/.{r->Inradius[t],s->Semiperimeter[t],Sequence@@Thread[{a,b,c}->SideLengths[Evaluate[t]]]}

Inner

CenterFunction[InnerSoddyPoint]

FullSimplify[Together//@(DistanceTrilinears[CenterFunction[InnerSoddyPoint]/.Cos[x_/2]:>Sqrt[(1+Cos[x])/2]/.Sec[x_/2]:>Sqrt[2/(1+Cos[x])]]),TriangleQ[]]

(1/2) (2+(1/(a Abs[ a-b-c]))(Sqrt[-(a+b-c) (a-b+c) (a+b+c)(a-b-c)]))/.(Sqrt[-(a+b-c) (a-b+c) (a+b+c)(a-b-c)])->a b c/R

alpha=1+((b c)/(2R (b+c-a)))

eqn=(1+((b c)/(2R (b+c-a)))/.R->Circumradius[])-(1/2) (2+(Sqrt[-(((a+b-c) (a-b+c) (a+b+c))/((a^2) (a-b-c)))]))

FullSimplify[Together//@eqn,TriangleQ[]]

rr=((4r^2 s)/(8r s+2(a b+b c+c a)-(a^2+b^2+c^2)));

rr/.s->Semiperimeter[]

Factor[Simplify[Together//@(DistanceTrilinears[CircleFunction[CircleEquation[CyclicTrilinears[alpha],rr]]]/.R->a b c/(4r s)/.s->Semiperimeter[]/.r^2->Inradius[]^2/.r^4->Inradius[]^4),TriangleQ[]]]

Factor[%/.{r^4->Inradius[]^4,r^3->r Inradius[]^2,r^2->Inradius[]^2}]

d1=((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)+(a^2) r-2 a b r+(b^2) r-2 a c r-2 b c r+(c^2) r);

{Factor[Expand[d1-Coefficient[d1,r]r]],Factor[-8Inradius[]^2 Semiperimeter[]]}

Coefficient[d1,r]

d2=((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)-4 a r-4 b r-4 c r);

Factor[Expand[d2-Coefficient[d2,r]r]]

Factor[{Coefficient[d2,r],-8Semiperimeter[]}]

f=Factor[-((-a+b-c) (a+b-c) (((-a+b+c))^3) (-15 (a^8)-52 (a^7) b+208 (a^6) (b^2)-20 (a^5) (b^3)-406 (a^4) (b^4)+324 (a^3) (b^5)+56 (a^2) (b^6)-124 a (b^7)+29 (b^8)-52 (a^7) c+96 (a^6) b c+276 (a^5) (b^2) c-424 (a^4) (b^3) c-396 (a^3) (b^4) c+624 (a^2) (b^5) c+44 a (b^6) c-168 (b^7) c+208 (a^6) (c^2)+276 (a^5) b (c^2)-388 (a^4) (b^2) (c^2)+72 (a^3) (b^3) (c^2)-952 (a^2) (b^4) (c^2)+612 a (b^5) (c^2)+172 (b^6) (c^2)-20 (a^5) (c^3)-424 (a^4) b (c^3)+72 (a^3) (b^2) (c^3)+544 (a^2) (b^3) (c^3)-532 a (b^4) (c^3)+360 (b^5) (c^3)-406 (a^4) (c^4)-396 (a^3) b (c^4)-952 (a^2) (b^2) (c^4)-532 a (b^3) (c^4)-786 (b^4) (c^4)+324 (a^3) (c^5)+624 (a^2) b (c^5)+612 a (b^2) (c^5)+360 (b^3) (c^5)+56 (a^2) (c^6)+44 a b (c^6)+172 (b^2) (c^6)-124 a (c^7)-168 b (c^7)+29 (c^8)-40 (a^7) r+88 (a^6) b r+216 (a^5) (b^2) r-488 (a^4) (b^3) r-56 (a^3) (b^4) r+456 (a^2) (b^5) r-120 a (b^6) r-56 (b^7) r+88 (a^6) c r+592 (a^5) b c r-24 (a^4) (b^2) c r-1696 (a^3) (b^3) c r+552 (a^2) (b^4) c r+592 a (b^5) c r-104 (b^6) c r+216 (a^5) (c^2) r-24 (a^4) b (c^2) r-592 (a^3) (b^2) (c^2) r-1008 (a^2) (b^3) (c^2) r+760 a (b^4) (c^2) r+648 (b^5) (c^2) r-488 (a^4) (c^3) r-1696 (a^3) b (c^3) r-1008 (a^2) (b^2) (c^3) r-2464 a (b^3) (c^3) r-488 (b^4) (c^3) r-56 (a^3) (c^4) r+552 (a^2) b (c^4) r+760 a (b^2) (c^4) r-488 (b^3) (c^4) r+456 (a^2) (c^5) r+592 a b (c^5) r+648 (b^2) (c^5) r-120 a (c^6) r-104 b (c^6) r-56 (c^7) r))/(16 b c (a+b+c)(((-8r s+((a^2+b^2+c^2)-2(a b+b c+a c))))^2) (((-8r^2 s+((a^2+b^2+c^2)-2(a b+b c+a c))r))^2))]

n=(-15 (a^8)-52 (a^7) b+208 (a^6) (b^2)-20 (a^5) (b^3)-406 (a^4) (b^4)+324 (a^3) (b^5)+56 (a^2) (b^6)-124 a (b^7)+29 (b^8)-52 (a^7) c+96 (a^6) b c+276 (a^5) (b^2) c-424 (a^4) (b^3) c-396 (a^3) (b^4) c+624 (a^2) (b^5) c+44 a (b^6) c-168 (b^7) c+208 (a^6) (c^2)+276 (a^5) b (c^2)-388 (a^4) (b^2) (c^2)+72 (a^3) (b^3) (c^2)-952 (a^2) (b^4) (c^2)+612 a (b^5) (c^2)+172 (b^6) (c^2)-20 (a^5) (c^3)-424 (a^4) b (c^3)+72 (a^3) (b^2) (c^3)+544 (a^2) (b^3) (c^3)-532 a (b^4) (c^3)+360 (b^5) (c^3)-406 (a^4) (c^4)-396 (a^3) b (c^4)-952 (a^2) (b^2) (c^4)-532 a (b^3) (c^4)-786 (b^4) (c^4)+324 (a^3) (c^5)+624 (a^2) b (c^5)+612 a (b^2) (c^5)+360 (b^3) (c^5)+56 (a^2) (c^6)+44 a b (c^6)+172 (b^2) (c^6)-124 a (c^7)-168 b (c^7)+29 (c^8)-40 (a^7) r+88 (a^6) b r+216 (a^5) (b^2) r-488 (a^4) (b^3) r-56 (a^3) (b^4) r+456 (a^2) (b^5) r-120 a (b^6) r-56 (b^7) r+88 (a^6) c r+592 (a^5) b c r-24 (a^4) (b^2) c r-1696 (a^3) (b^3) c r+552 (a^2) (b^4) c r+592 a (b^5) c r-104 (b^6) c r+216 (a^5) (c^2) r-24 (a^4) b (c^2) r-592 (a^3) (b^2) (c^2) r-1008 (a^2) (b^3) (c^2) r+760 a (b^4) (c^2) r+648 (b^5) (c^2) r-488 (a^4) (c^3) r-1696 (a^3) b (c^3) r-1008 (a^2) (b^2) (c^3) r-2464 a (b^3) (c^3) r-488 (b^4) (c^3) r-56 (a^3) (c^4) r+552 (a^2) b (c^4) r+760 a (b^2) (c^4) r-488 (b^3) (c^4) r+456 (a^2) (c^5) r+592 a b (c^5) r+648 (b^2) (c^5) r-120 a (c^6) r-104 b (c^6) r-56 (c^7) r);

eqns=Numerator/@Together//@{r^2-Inradius[]^2,s-Semiperimeter[]}

Factor[GroebnerBasis[{n,Sequence@@eqns},{a,b,c},MonomialOrder->EliminationOrder]]

Factor[Factor[(b^2) c+b (c^2)-(b^2) s-3 b c s-(c^2) s-(r^2) s+2 b (s^2)+2 c (s^2)-(s^3)/.s->Semiperimeter[]]/.r^2->Inradius[]^2]

n1=Factor[n-Coefficient[n,r]r]

nr=Factor[Coefficient[n,r]]

Factor[GroebnerBasis[{n1,Sequence@@eqns},{a,b,c},MonomialOrder->EliminationOrder]]

Factor[GroebnerBasis[{(5 (a^6)-16 (a^5) b-11 (a^4) (b^2)+72 (a^3) (b^3)-65 (a^2) (b^4)+8 a (b^5)+7 (b^6)-16 (a^5) c-42 (a^4) b c+56 (a^3) (b^2) c+84 (a^2) (b^3) c-88 a (b^4) c+6 (b^5) c-11 (a^4) (c^2)+56 (a^3) b (c^2)-38 (a^2) (b^2) (c^2)+80 a (b^3) (c^2)-87 (b^4) (c^2)+72 (a^3) (c^3)+84 (a^2) b (c^3)+80 a (b^2) (c^3)+148 (b^3) (c^3)-65 (a^2) (c^4)-88 a b (c^4)-87 (b^2) (c^4)+8 a (c^5)+6 b (c^5)+7 (c^6)),Sequence@@eqns},{a,b,c},MonomialOrder->EliminationOrder]]

n1

nr

nrr= (5 (a^6)-16 (a^5) b-11 (a^4) (b^2)+72 (a^3) (b^3)-65 (a^2) (b^4)+8 a (b^5)+7 (b^6)-16 (a^5) c-42 (a^4) b c+56 (a^3) (b^2) c+84 (a^2) (b^3) c-88 a (b^4) c+6 (b^5) c-11 (a^4) (c^2)+56 (a^3) b (c^2)-38 (a^2) (b^2) (c^2)+80 a (b^3) (c^2)-87 (b^4) (c^2)+72 (a^3) (c^3)+84 (a^2) b (c^3)+80 a (b^2) (c^3)+148 (b^3) (c^3)-65 (a^2) (c^4)-88 a b (c^4)-87 (b^2) (c^4)+8 a (c^5)+6 b (c^5)+7 (c^6));

f2=-((-a+b-c) (a+b-c) (((-a+b+c))^3) (n1-16s r nrr))/(16 b c (a+b+c) (r^2) ((((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)-8 r s))^4))

f1=Factor[-((-a+b-c) (a+b-c) (((-a+b+c))^3) (-15 (a^8)-52 (a^7) b+208 (a^6) (b^2)-20 (a^5) (b^3)-406 (a^4) (b^4)+324 (a^3) (b^5)+56 (a^2) (b^6)-124 a (b^7)+29 (b^8)-52 (a^7) c+96 (a^6) b c+276 (a^5) (b^2) c-424 (a^4) (b^3) c-396 (a^3) (b^4) c+624 (a^2) (b^5) c+44 a (b^6) c-168 (b^7) c+208 (a^6) (c^2)+276 (a^5) b (c^2)-388 (a^4) (b^2) (c^2)+72 (a^3) (b^3) (c^2)-952 (a^2) (b^4) (c^2)+612 a (b^5) (c^2)+172 (b^6) (c^2)-20 (a^5) (c^3)-424 (a^4) b (c^3)+72 (a^3) (b^2) (c^3)+544 (a^2) (b^3) (c^3)-532 a (b^4) (c^3)+360 (b^5) (c^3)-406 (a^4) (c^4)-396 (a^3) b (c^4)-952 (a^2) (b^2) (c^4)-532 a (b^3) (c^4)-786 (b^4) (c^4)+324 (a^3) (c^5)+624 (a^2) b (c^5)+612 a (b^2) (c^5)+360 (b^3) (c^5)+56 (a^2) (c^6)+44 a b (c^6)+172 (b^2) (c^6)-124 a (c^7)-168 b (c^7)+29 (c^8)-16 (5 (a^6)-16 (a^5) b-11 (a^4) (b^2)+72 (a^3) (b^3)-65 (a^2) (b^4)+8 a (b^5)+7 (b^6)-16 (a^5) c-42 (a^4) b c+56 (a^3) (b^2) c+84 (a^2) (b^3) c-88 a (b^4) c+6 (b^5) c-11 (a^4) (c^2)+56 (a^3) b (c^2)-38 (a^2) (b^2) (c^2)+80 a (b^3) (c^2)-87 (b^4) (c^2)+72 (a^3) (c^3)+84 (a^2) b (c^3)+80 a (b^2) (c^3)+148 (b^3) (c^3)-65 (a^2) (c^4)-88 a b (c^4)-87 (b^2) (c^4)+8 a (c^5)+6 b (c^5)+7 (c^6)) r s))/(16 b c (a+b+c) (r^2) ((((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)-8 r s))^4))/.r^-2->Inradius[]^-2]

Expand[n-(n1-16 s r nrr)/.s->Semiperimeter[]]

Factor[f-f1/.s->Semiperimeter[]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[InnerSoddyCenter]]]

Outer

Same with r s terms having flipped signs

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[OuterSoddyCenter]]]

Circle Centers

IncidentCenters[KimberlingCenters,CircleCenter[InnerSoddyCircle]]

CenterName@@%

IncidentCenters[KimberlingCenters,CircleCenter[OuterSoddyCircle]]

CenterName@@%

Points

IncidentCenters[KimberlingCenters,InnerSoddyCircle]

IncidentCenters[KimberlingCenters,OuterSoddyCircle]

Bellew's Generalization

bellew=(((n(c-1)^2+1)Sum[(Subscript[κ, i]),{i,n+1}]+n c(c-1)(n c^2+(3-n)c-4))^2)/(n(c-1)+1)^2-
(n(c-1)^2+1)Sum[(Subscript[κ, i])^2,{i,n+1}]+n(3n c^2-2n-6)c^2(c-1)^2

FullSimplify[bellew]

FullSimplify[bellew /. c->Csc[Pi/n]]

FullSimplify[bellew /. c->Csc[Pi/n]/. n->3]

FullSimplify[bellew /. c->Csc[Pi/n]/. n->4]