Outer Napoleon Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Equilateral

EquilateralQ[OuterNapoleonTriangle[RandomTriangle[20]]]

Simplify[#^2&/@EquilateralQ[OuterNapoleonTriangle]/.C->π-A-B]

Figure

V5

t=Triangle[{{0.,0.},{2.,-.3},{.9,.7}}];
Show[
Graphics[{
{Thickness[.01],t},
{Purple,nt=OuterNapoleonTriangle[t],
EqualSideTicks[nt,4,.04,.03]
},
Transpose[{{Red,Green,Blue},e=ErectPolygons[t,1],MapIndexed[EqualSideTicks[#1,#2[[1]],.04,.02]&,e]}],
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Purple]&/@Transpose[{Point/@Vertices[nt],{"(Subscript[N, A])","(Subscript[N, B])","(Subscript[N, C])"}}],
Transpose[{Point/@(#[[1]]&/@Vertices/@e),{"(Subscript[E, A])","(Subscript[E, B])","(Subscript[E, C])"},{Red,Green,Blue}}]
],{PointSize[.025],LabelOffset->.08}
]
}],AspectRatio->Automatic,PlotRange->All]

Updated by hand

Import["/Users/eww/Documents/Troves/MathOzTeX/graphics/eps/OuterNapoleonTriangle.eps"]

Trilinears

Trilinears[OuterNapoleonTriangle]//MatrixForm//TraditionalForm

Side Lengths

abc=Together//@DistanceTrilinears[TrigExpand//@SideLengths[OuterNapoleonTriangle]/.{Cos[x_+y_]:>TrigExpand[Cos[x+y]],Cos[x_-y_]:>TrigExpand[Cos[x-y]],
Sin[x_+y_]:>TrigExpand[Sin[x+y]],Sin[x_-y_]:>TrigExpand[Sin[x-y]]}]

FullSimplify[abc,TriangleQ[]]

Area

area=Simplify[Together//@DistanceTrilinears[TrigExpand[Area[OuterNapoleonTriangle]]],#>0&/@{a,b,c}]

FullSimplify[(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-4Area[],TriangleQ[]]

FullSimplify[(1/24) ((Sqrt[3]) (a^2)+(Sqrt[3]) (b^2)+(Sqrt[3]) (c^2)+12Δ)]

Δ/2+Sqrt[3](a^2+b^2+c^2)/24

Centers

(pts={#,KimberlingLookup[#[OuterNapoleonTriangle]]}&/@Take[KimberlingCenters,10]);//Timing

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm