Mixtilinear Incircles

Author

Eric W. Weisstein
January 28, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Graphic

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
Circumcircle[t],
Transpose[{{Red,Blue,Green},MixtilinearIncircles[t]}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Transpose[{Center/@MixtilinearIncircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"},{Red,Blue,Green}}],
{
{Circumcenter[t],"O"}
}
],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Derivation

d2=Simplify[Factor[Together//@ToSidelengths[Distance[Trilinears[Circumcenter],{α,β,γ}]]]^2,TriangleQ[]]

Factor[d2]//TraditionalForm

R=Circumradius[](((1/4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]))/area)

(4area)^2 Numerator[d2]-Denominator[d2](a b c-4 area r)^2

Distance[{α,β,γ},TrilinearLine[{0,1,0}]](area/((1/4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])))

Distance[{α,β,γ},TrilinearLine[{0,0,1}]](area/((1/4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])))

Area[]^2

(2(a α+b β+c γ))^2 Numerator[d2]-Denominator[d2](a b c-2(a α+b β+c γ)r)^2

(2Area[])^2==(a α+b β+c γ)^2

FullSimplify[{
(2(a α+b β+c γ))^2 Numerator[d2]-Denominator[d2](a b c-2(a α+b β+c γ)r)^2==0,
4(((a α+b β+c γ))^2)-(a+b-c) (a-b+c) (-a+b+c) (a+b+c)==0}/.{β->r,γ->r}]

Solve

soln=Solve[{((((b+c) r+a α))^2) (-4 a b c (a ((a^4)+(b^4)+(b^3) c+b (c^3)+(c^4)-2 (a^2) ((b^2)+(c^2))) (r^2)+(b+c) ((a^4)+(((b-c))^2) (((b+c))^2)-2 (a^2) ((b^2)-b c+(c^2))) r α+(a^3) b c (α^2))-(a-b-c) (a+b-c) (a-b+c) (a+b+c) (((a b c-2 r ((b+c) r+a α)))^2))==0,4 ((((b+c) r+a α))^2)==(a+b-c) (a-b+c) (-a+b+c) (a+b+c)},{r,α}]

r1=FullSimplify[r/.soln[[-1]],TriangleQ[]]

r2=Factor[FullSimplify[Factor[ToSidelengths[HalfAngleFormulas[Inradius[]Sec[A/2]^2]]],TriangleQ[]]]

FullSimplify[r1-r2,TriangleQ[]]

alpha=Together//@(α/.soln[[-1]])

soln/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Factor[Refine[Map[Factor,alpha,{2}],TriangleQ[]]]

-(Sqrt[(((a+b-c) (a-b+c))/( (-a+b+c)))]) ((((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)))/(2 a  (((a+b+c))^(3/2))))//InputForm

Inradius[] Sec[A/2]^2 /.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]

Factor[ToSidelengths[Inradius[] HalfAngleFormulas[Sec[A/2]^2]]]

-(Sqrt[(((a+b-c) (a-b+c))/( (-a+b+c)))]) ((((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)))/(2 a  (((a+b+c))^(3/2))))/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

-Inradius[] Sec[A/2]^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)))/(4a b c ))/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]

ToAngles[((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))]

(1/2)Inradius[] Sec[A/2]^2( 1+Cos[A]-Cos[B]-Cos[C])/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]

Groebner Basis

rgb=Factor/@GroebnerBasis[{
((((b+c) r+a α))^2) (-4 a b c (a ((a^4)+(b^4)+(b^3) c+b (c^3)+(c^4)-2 (a^2) ((b^2)+(c^2))) (r^2)+(b+c) ((a^4)+(((b-c))^2) (((b+c))^2)-2 (a^2) ((b^2)-b c+(c^2))) r α+(a^3) b c (α^2))-(a-b-c) (a+b-c) (a-b+c) (a+b+c) (((a b c-2 r ((b+c) r+a α)))^2)),4 ((((b+c) r+a α))^2)-(a+b-c) (a-b+c) (-a+b+c) (a+b+c)
},r,α,MonomialOrder->EliminationOrder]

Length[rgb]

αgb=Factor/@GroebnerBasis[{
((((b+c) r+a α))^2) (-4 a b c (a ((a^4)+(b^4)+(b^3) c+b (c^3)+(c^4)-2 (a^2) ((b^2)+(c^2))) (r^2)+(b+c) ((a^4)+(((b-c))^2) (((b+c))^2)-2 (a^2) ((b^2)-b c+(c^2))) r α+(a^3) b c (α^2))-(a-b-c) (a+b-c) (a-b+c) (a+b+c) (((a b c-2 r ((b+c) r+a α)))^2)),4 ((((b+c) r+a α))^2)-(a+b-c) (a-b+c) (-a+b+c) (a+b+c)
},α,r,MonomialOrder->EliminationOrder]

Length[αgb]