Outer Napoleon Circle

Author

Eric W. Weisstein
February 15, 2006

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

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

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

Figure

t=Triangle[{{0,0},{2,-3/10},{9/10,7/10}}];
Show[
Graphics[{
{Thickness[.01],t},
{Purple,nt=OuterNapoleonTriangle[t],
EqualSideTicks[nt,4,.04,.06]
},
Transpose[{{Yellow,Green,Blue},e=ErectPolygons[t,1],MapIndexed[EqualSideTicks[#1,#2[[1]],.04,.02]&,e]}],
{Red,OuterNapoleonCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[nt],{"(Subscript[N, A])","(Subscript[N, B])","(Subscript[N, C])"}}],
Transpose[{Point/@(#[[1]]&/@Vertices/@e),{"(Subscript[E, A])","(Subscript[E, B])","(Subscript[E, C])"},{Yellow,Green,Blue}}],
{
{Centroid[t],"G",Red}
}
],{PointSize[.025],LabelOffset->.08}
]
}],AspectRatio->Automatic,PlotRange->All]

Center

Center[OuterNapoleonCircle]

IncidentCenters[{X[2]},Center[OuterNapoleonCircle]]//Timing

CenterName[X[2]]

Radius

CircleRadius[OuterNapoleonCircle]//TraditionalForm

Circle Function

TraditionalForm[l=CircleFunction[OuterNapoleonCircle]]

IncidentCenters[{X[16]},CyclicTrilinears[l]]//Timing

CenterName[X[16]]

CenterFunction[X[16]]

Incident Centers

(ctr=IncidentCenters[KimberlingCenters,OuterNapoleonCircle])//Timing

{#,Name[#]}&/@ctr//ColumnForm

Inverse Points

(inv=Union[Sort/@Flatten/@Select[{#[[1]],KimberlingLookup[InversePoint[#,OuterNapoleonCircle]]}&/@Take[KimberlingCenters,All],Last[#]≠{}&]])//Timing

{X[#[[1]]],Name[X[#[[1]]]],X[#[[2]]],Name[X[#[[2]]]]}&/@DeleteCases[inv,{2,_}]//TableForm

Derivation

TrigExpand[TrilinearVertexMatrix[OuterNapoleonTriangle]]

eqn=Together//@ToSidelengths[TrilinearEquation@@(TrilinearVertexMatrix[OuterNapoleonTriangle]/.Sin[x_]:>TrigExpand[Sin[x]])]

eqn2=Factor[eqn/.Sqrt[x_]:>Simplify[Sqrt[x],TriangleQ[]]]

eqn3=Factor[eqn2/.{(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])->s,((((a+b-c) (a-b+c) (-a+b+c) (a+b+c)))^(3/2))->s(a+b-c) (a-b+c) (-a+b+c) (a+b+c)}]

eqn=(2 (a^8) (α^2)-5 (a^6) (b^2) (α^2)+4 (a^4) (b^4) (α^2)-(a^2) (b^6) (α^2)-5 (a^6) (c^2) (α^2)+(a^2) (b^4) (c^2) (α^2)+4 (a^4) (c^4) (α^2)+(a^2) (b^2) (c^4) (α^2)-(a^2) (c^6) (α^2)-(Sqrt[3]) (a^4) (b^2) s (α^2)+(Sqrt[3]) (a^2) (b^4) s (α^2)-(Sqrt[3]) (a^4) (c^2) s (α^2)+(Sqrt[3]) (a^2) (c^4) s (α^2)+(a^7) b α β-(a^5) (b^3) α β-(a^3) (b^5) α β+a (b^7) α β+5 (a^5) b (c^2) α β-18 (a^3) (b^3) (c^2) α β+5 a (b^5) (c^2) α β-13 (a^3) b (c^4) α β-13 a (b^3) (c^4) α β+7 a b (c^6) α β+(Sqrt[3]) (a^5) b s α β-2 (Sqrt[3]) (a^3) (b^3) s α β+(Sqrt[3]) a (b^5) s α β-4 (Sqrt[3]) (a^3) b (c^2) s α β-4 (Sqrt[3]) a (b^3) (c^2) s α β-(Sqrt[3]) a b (c^4) s α β-(a^6) (b^2) (β^2)+4 (a^4) (b^4) (β^2)-5 (a^2) (b^6) (β^2)+2 (b^8) (β^2)+(a^4) (b^2) (c^2) (β^2)-5 (b^6) (c^2) (β^2)+(a^2) (b^2) (c^4) (β^2)+4 (b^4) (c^4) (β^2)-(b^2) (c^6) (β^2)+(Sqrt[3]) (a^4) (b^2) s (β^2)-(Sqrt[3]) (a^2) (b^4) s (β^2)-(Sqrt[3]) (b^4) (c^2) s (β^2)+(Sqrt[3]) (b^2) (c^4) s (β^2)+(a^7) c α γ+5 (a^5) (b^2) c α γ-13 (a^3) (b^4) c α γ+7 a (b^6) c α γ-(a^5) (c^3) α γ-18 (a^3) (b^2) (c^3) α γ-13 a (b^4) (c^3) α γ-(a^3) (c^5) α γ+5 a (b^2) (c^5) α γ+a (c^7) α γ+(Sqrt[3]) (a^5) c s α γ-4 (Sqrt[3]) (a^3) (b^2) c s α γ-(Sqrt[3]) a (b^4) c s α γ-2 (Sqrt[3]) (a^3) (c^3) s α γ-4 (Sqrt[3]) a (b^2) (c^3) s α γ+(Sqrt[3]) a (c^5) s α γ+7 (a^6) b c β γ-13 (a^4) (b^3) c β γ+5 (a^2) (b^5) c β γ+(b^7) c β γ-13 (a^4) b (c^3) β γ-18 (a^2) (b^3) (c^3) β γ-(b^5) (c^3) β γ+5 (a^2) b (c^5) β γ-(b^3) (c^5) β γ+b (c^7) β γ-(Sqrt[3]) (a^4) b c s β γ-4 (Sqrt[3]) (a^2) (b^3) c s β γ+(Sqrt[3]) (b^5) c s β γ-4 (Sqrt[3]) (a^2) b (c^3) s β γ-2 (Sqrt[3]) (b^3) (c^3) s β γ+(Sqrt[3]) b (c^5) s β γ-(a^6) (c^2) (γ^2)+(a^4) (b^2) (c^2) (γ^2)+(a^2) (b^4) (c^2) (γ^2)-(b^6) (c^2) (γ^2)+4 (a^4) (c^4) (γ^2)+4 (b^4) (c^4) (γ^2)-5 (a^2) (c^6) (γ^2)-5 (b^2) (c^6) (γ^2)+2 (c^8) (γ^2)+(Sqrt[3]) (a^4) (c^2) s (γ^2)+(Sqrt[3]) (b^4) (c^2) s (γ^2)-(Sqrt[3]) (a^2) (c^4) s (γ^2)-(Sqrt[3]) (b^2) (c^4) s (γ^2));

Center

ctr=-Symmetrize[Factor[ExpandAll[Center[eqn][[1]]/.s->(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]]]

IncidentCenters[Take[KimberlingCenters,10],1/{a,b,c}]

CenterName[X[2]]

Circle Function

Simplify[CircleFunction[Circle[OuterNapoleonCircle]]]

Radius

FullSimplify[Factor[Together//@ToSidelengths[Distance[Center[OuterNapoleonCircle],TrilinearVertexMatrix[OuterNapoleonTriangle][[1]]]/.f:_Sin:>TrigExpand[f]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>Refine[Sqrt[x],TriangleQ[]]],TriangleQ[]]

((Sqrt[(a^2)+(b^2)+(c^2)+(Sqrt[3]) 4Area])/(3 (Sqrt[2])))//TraditionalForm