Euler Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.4,.7}}];
Show[Graphics[{
t,
EqualSideTicksAboutMidpoints[t,.015,.012],
{Red,EulerTriangle[t]},
{Blue,Altitudes[t],
MapIndexed[EqualSideTicksAboutMidpoint[#1,Sequence@@#2,.015,.03]&,
Line[{Coordinates[Orthocenter[t]],#}]&/@Vertices[t]]
},
{Purple,NinePointCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{EulerPoints[t],{"(Subscript[E, A])","(Subscript[E, B])","(Subscript[E, C])"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[OrthicTriangle[t]],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
{
{Orthocenter[t],"H",Blue}
}
],{PointSize[.02],LabelOffset->.03}]
}],AspectRatio->Automatic,PlotRange->All]

Similarity

SimilarQ[t,EulerTriangle[t]]

Center and Radius

rs=Factor[Refine[Together//@(ToSidelengths[Distance[#,TrigExpand/@Trilinears[NinePointCenter]]^2&/@TrilinearVertexMatrix[EulerTriangle]])/.Sqrt[x_]Sqrt[y_]:>Refine[Sqrt[x y],TriangleQ[]],TriangleQ[]]]

Refine[Sqrt[rs],TriangleQ[]]

Equal@@%

CircleRadius[NinePointCircle]

Centers

(pts={#,KimberlingLookup[#[EulerTriangle]]}&/@Take[KimberlingCenters,1000]);//Timing

l=Map[First,DeleteCases[pts,_?(Last[#]==={}&)],{2}]

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

(pts={#,KimberlingLookup[#[EulerTriangle]]}&/@Take[KimberlingCenters,{1001,1500}]);//Timing

l=Map[First,DeleteCases[pts,_?(Last[#]==={}&)],{2}]

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

...crash...