Inner Napoleon Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Equilateral

EquilateralQ[InnerNapoleonTriangle[RandomTriangle[20]]]

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

Figure

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

Trilinears

Trilinears[InnerNapoleonTriangle]//MatrixForm//TraditionalForm

Area

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

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

Δ/2-Sqrt[3](a^2+b^2+c^2)/24

Centers

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

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