Anticomplementary Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{
Red,
at=AnticevianTriangle[t,Centroid[t]],
EqualSideTicksAboutMidpoints[at,.03,.015]
},
{
Blue,
Line/@Transpose[Vertices/@{t,at}]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,Red]&/@Transpose[{ExmedianPoints[t],{"A(Subscript[', 1])","A(Subscript[', 2])","A(Subscript[', 3])"}}],
{{Centroid[t],"G",Blue}}
],
{PointSize[.025],LabelOffset->.06}
]
}],AspectRatio->Automatic,PlotRange->All]

Vertices

AnticevianTrilinears[Centroid]//MatrixForm//TraditionalForm

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

Similarity

SimilarQ[t,at]

Side Lengths

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

Area

Timing[area=FullSimplify[DistanceTrilinears[AnticevianTriangleArea[Centroid]],TriangleQ[]] Δ/TriangleArea[]]

Centers

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

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