Excircles Radical Circle

Author

Eric W. Weisstein
December 9, 2004

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{3/10,5/10}}];
excircles=Through[Excircles[t]];
nrc=RadicalCenter[excircles];
nr=Sqrt[((Distance[nrc,CircleCenter[excircles[[1]]]])^2)-((CircleRadius[excircles[[1]]])^2)];
Show[Graphics[{
{Purple,Sides[t,10]},
t,
{Green,RadicalLines[excircles,5]},
{Blue,excircles},
{Red,Circle[Coordinates[nrc],nr]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{CircleCenter/@excircles,{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
{{nrc,"Sp",Red}}
]
]
}],AspectRatio->Automatic,PlotRange->{{-1,2},{-1.5,1.5}}]

Orthogonality Check

OrthogonalQ[N[Circle[Coordinates[nrc],nr],30],#]&/@excircles

Derivation

eqns0=Factor/@(Simplify[Together//@ToSidelengths[CircleEquation[#]],TriangleQ[]]&/@Excircles)

eqns=((16 (a^2) (b^2) (c^2))/((a-b-c) (a+b-c) (a-b+c) (a+b+c) ))eqns0

(ls=CircleFunction/@eqns)//TraditionalForm

CircleFunction[eqn=(a^4) (α^2)+2 (a^3) b (α^2)+(a^2) (b^2) (α^2)+2 (a^3) c (α^2)+2 (a^2) b c (α^2)+(a^2) (c^2) (α^2)+2 (a^3) b α β+4 (a^2) (b^2) α β+2 a (b^3) α β-2 a b (c^2) α β+(a^2) (b^2) (β^2)+2 a (b^3) (β^2)+(b^4) (β^2)-2 a (b^2) c (β^2)-2 (b^3) c (β^2)+(b^2) (c^2) (β^2)+2 (a^3) c α γ-2 a (b^2) c α γ+4 (a^2) (c^2) α γ+2 a (c^3) α γ-2 (a^2) b c β γ+2 (b^3) c β γ-4 (b^2) (c^2) β γ+2 b (c^3) β γ+(a^2) (c^2) (γ^2)-2 a b (c^2) (γ^2)+(b^2) (c^2) (γ^2)+2 a (c^3) (γ^2)-2 b (c^3) (γ^2)+(c^4) (γ^2)]

Factor/@Coefficient[eqn,#,2]&/@{α,β,γ}

CircleFunction[eqn_]:=Module[{k,l,m,n,soln},soln=Solve[{c k+b l+a m==Coefficient[eqn,α β],b k+c l+a n==Coefficient[eqn,α γ],a k+c m+b n==Coefficient[eqn,β γ],
a l==Coefficient[eqn,α^2],
b m==Coefficient[eqn,β^2],
c n==Coefficient[eqn,γ^2]
},{l,m,n,k}][[1]];
{l,m,n}/k/.soln]

rcenter=Symmetrize/@Factor[RadicalCenter[ls,{1,1,1}]]

nrc//N

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

r=Factor[Together//@Sqrt[((Distance[rcenter,CircleCenter[Excircles[[1]]]])^2)-((CircleRadius[Excircles[[1]]])^2)]]

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

nr//N

Simplify[Together//@ToSidelengths[CircleEquation[Circle[{((b+c)/a),((a+c)/b),((a+b)/c)},(1/2) (Sqrt[(((a^2) b+a (b^2)+(a^2) c+a b c+(b^2) c+a (c^2)+b (c^2))/(a+b+c))])]]],TriangleQ[]]

Factor[%]

CircleFunction[((a^4) (α^2)-(a^2) (b^2) (α^2)+2 (a^2) b c (α^2)-(a^2) (c^2) (α^2)+2 (a^2) b c α β+2 a (b^2) c α β+2 a b (c^2) α β-(a^2) (b^2) (β^2)+(b^4) (β^2)+2 a (b^2) c (β^2)-(b^2) (c^2) (β^2)+2 (a^2) b c α γ+2 a (b^2) c α γ+2 a b (c^2) α γ+2 (a^2) b c β γ+2 a (b^2) c β γ+2 a b (c^2) β γ-(a^2) (c^2) (γ^2)+2 a b (c^2) (γ^2)-(b^2) (c^2) (γ^2)+(c^4) (γ^2))]

CircleCenterFunction[ExcirclesRadicalCircle]:=(b+c)/a
CircleFunction[ExcirclesRadicalCircle]:=(a^3-a b^2+2 a b c-a c^2)/(4 a b c)
CircleRadius[ExcirclesRadicalCircle]:= Sqrt[(a^2 b+a b^2+a^2 c+a b c+b^2 c+a c^2+b c^2)/(a+b+c)]/2

Circle Center

CircleCenter[ExcirclesRadicalCircle]

IncidentCenters[KimberlingCenters,CircleCenter[ExcirclesRadicalCircle]]//Timing

CenterName[X[10]]

Circle Function

CircleFunction[ExcirclesRadicalCircle]

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

CenterName[X[56]]

Circle Radius

CircleRadius[ExcirclesRadicalCircle]

Incident Centers

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