Medial Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.4,.7}}];
Show[Graphics[{
t,
{Red,mt=MedialTriangle[t]},
{Green,Cevians[t,Centroid[t]]},
{
Blue,
p=PedalLines[t,Circumcenter[t]],
PerpSquare[#,.03,2]&/@Transpose[{Sides[t],p}]
},
EqualSideTicksAboutMidpoints[t,.02,.02],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[mt],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
{
{Circumcenter[t],"O",Blue},
{Centroid[t],"G",Green}
}
],{PointSize[.025],LabelOffset->.04}]
}],AspectRatio->Automatic,PlotRange->All]

Vertices

CevianTrilinears[Centroid]//MatrixForm//TraditionalForm

a b c CevianTrilinears[Centroid]//MatrixForm//TraditionalForm

Similarity

SimilarQ[t,mt]

Side Lengths

abc=FullSimplify[DistanceTrilinears[CevianTriangleSideLengths[Centroid]],TriangleQ[]]

Area

CevianTriangleArea[Centroid]Δ/TriangleArea[]

Centers

(pts={#,KimberlingLookup[#[MedialTriangle]]}&/@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