Steiner Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{3/10,5/10}}]//N[#,10]&;
Show[Graphics[
{
{Green,Sides[t,2]},
t,
{Blue,
(Cartesian[#[t]]&/@(Cevians[SteinerPoint]/.Line[x__]:>Line[x,Trilinears[SteinerPoint]]))/.Line:>LongestLine
},
{Red,SteinerTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[SteinerTriangle[t]],{"(Subscript[S, A])","(Subscript[S, B])","(Subscript[S, C])"}}],
{
{SteinerPoint[t],"S",Blue}
}
],LabelOffset->.06,PointSize[.025]]
}
],AspectRatio->Automatic,PlotRange->{{-.2,2},{-.2,1.5}}]

Trilinear Vertex Matrix

Vertices[Triangle[SteinerTriangle]]//TraditionalForm

Perspectrix

i=Intersections@@@Partition[Line@@@Transpose[TrilinearVertexMatrix/@{SteinerTriangle,ReferenceTriangle}],2,1,1]

-Map[Symmetrize,ExactTrilinears/@i,2]

Symmetrize/@Trilinears[SteinerPoint]

Area

area=Factor[Together//@Area[SteinerTriangle]]

Simplify[area(Δ/(Area[])),TriangleQ[]]

Side Lengths

Map[Factor,Simplify[Map[Factor,Together//@ToSidelengths[SideLengths[SteinerTriangle]],3],TriangleQ[]],3]

Centers

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

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