Inner Soddy Circle

Author

Eric W. Weisstein
November 15, 2005

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/InnerSoddyCircle.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]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{InnerSoddyCenter[t],"S",Red}
}
],PointSize[.022]]
},AspectRatio->Automatic]
]

Incident Points

<<MathWorld KimberlingCenters`

(ctrs=IncidentCenters[KimberlingCenters,InnerSoddyCircle])//Timing

{#,CenterName[#]}&/@ctrs//TableForm

Circle Function

CircleFunction[InnerSoddyCircle]

l1=((((-a+b+c))^2) (-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)-4 (Sqrt[(((a+b-c) (a-b+c) (-a+b+c))/(a+b+c))]) (a+b+c) (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))))/(4 b c ((((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)-2 (Sqrt[(((a+b-c) (a-b+c) (-a+b+c))/(a+b+c))]) (a+b+c)))^4));

Inradius[]

Factor[CircleFunction[InnerSoddyCircle]/.(Sqrt[(((a+b-c) (a-b+c) (-a+b+c))/(a+b+c))]):>2r]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[InnerSoddyCircle]]]//Timing

l2=With[{rA=4Circumradius Sin[A/2]Cos[B/2]Cos[C/2],rB=4Circumradius Cos[A/2]Sin[B/2]Cos[C/2],rC=4Circumradius Cos[A/2]Cos[B/2]Sin[C/2]},-((c^2 (b+rB)^2+b^2 (c+rC)^2+2 (b+rB) (c+rC) SA)/(rA+rB+rC+2 Semiperimeter)^2-(S^2/(4 Semiperimeter (S (Cot[BrocardAngle]-1)-Semiperimeter^2)))^2)/(b c)];

t=RandomTriangle[20];

{l1,l2}/.TriangleRules/.Thread[{A,B,C}->Angles[t]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

-((c^2 (b+rB)^2+b^2 (c+rC)^2+2 (b+rB) (c+rC) SA)/(rA+rB+rC+2 Semiperimeter)^2-(S^2/(4 Semiperimeter (S (Cot[BrocardAngle]-1)-Semiperimeter^2)))^2)/(b c)//TraditionalForm

(1/(b c))(((S^4)/(16 (s^2) (((S (cot(ω)-1)-(s^2)))^2)))-(((c^2) (((b+rB))^2)+2 (c+rC) (Subscript[S, A]) (b+rB)+(b^2) (((c+rC))^2))/(((rA+rB+rC+2 s))^2)))

Center

CircleCenter[InnerSoddyCircle]

Radius

CircleRadius[InnerSoddyCircle]