Lemoine Triangle

Author

Eric W. Weisstein
November 11, 2005

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LemoineTriangle.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[
{
t,
{
Blue,
LemoineTriangle[t],
Cartesian/@Through[Cevians[X[598]][t]]
},
{Red,LemoineTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[LemoineTriangle[t]],{"(Subscript[X, A])","(Subscript[X, B])","(Subscript[X, C])"}}],
{
{X[598][t],"(Subscript[X, 598])",Blue}
}
],LabelOffset->.04,PointSize[.025]]
}
],AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.1,.6}}]

Trilinear Vertex Matrix

Vertices[Triangle[LemoineTriangle]]//TraditionalForm

CenterName[X[598]]

CenterName[X[574]]

Perspectrix

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

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

Symmetrize/@Trilinears[X[598]]

Area

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

Side Lengths

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

Centers

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

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