Symmedial Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[
{
t,
{Blue,Symmedians[t]},
{Red,SymmedialTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[SymmedialTriangle[t]],{"(Subscript[K, A])","(Subscript[K, B])","(Subscript[K, C])"}}],
{
{SymmedianPoint[t],"K",{Blue,LabelOffset->{-.035,.05}}}
}
],LabelOffset->{.05,-.02},PointSize[.025]]
}
],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

Vertices[Triangle[SymmedialTriangle]]//TraditionalForm

Perspectrix

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

Map[Symmetrize,ExactTrilinears/@i,2]

Trilinears[SymmedianPoint]

Area

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

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

Given by Casey, p. 172, apparently incorrectly

FullSimplify[ToSidelengths[((12(Area^3))/((((a^2)+(b^2)+(c^2)))^2))],TriangleQ[]]

Side Lengths

Map[Factor,Simplify[Together//@ToSidelengths[SideLengths[SymmedialTriangle]],TriangleQ[]],2]

Centers

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

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

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