Anticevian Triangle

Author

Eric W. Weisstein
January 21, 2005

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/AnticevianTriangle.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,alpha={.3,.3,.2}]},
{Blue,Anticevians[t,alpha]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[at],{"A'","B'","C'"}}],
{{Point[TrilinearToCartesian[t,alpha]],"P",Blue}}
],{PointSize[.02],LabelOffset->.07}]
}],AspectRatio->Automatic,PlotRange->All]

Side Lengths

Timing[l=FullSimplify[AnticevianTriangleSideLengths[{α,β,γ}][[1]],TriangleQ[]&&0<#<1&/@{α,β,γ}]]

l1=Expand[ (-(b^2) β γ+(a-c) (a+c) β γ+b c ((β^2)+(γ^2)))]

l2=b c((β^2)+(γ^2))- β γ(-a^2+b^2+c^2)

Simplify[l1-l2]

Factor[LawOfCosines[l2]]

l3=((2 a α (Sqrt[b c (-(b^2) β γ+(a-c) (a+c) β γ+b c ((β^2)+(γ^2)))]))/(Abs[a α+b β-c γ] Abs[a α-b β+c γ]));

abc=CyclicTrilinears[((2 a b c α (Sqrt[(β^2)+(γ^2)-2 β γ Cos[A]]))/(Abs[(b β-c γ)^2-(a α)^2] ))]

Sum

FullSimplify[Plus@@abc,TriangleQ[]&&0<#<1&/@{α,β,γ}]

Product

FullSimplify[Times@@abc,TriangleQ[]&&0<#<1&/@{α,β,γ}]

Area

FullSimplify[Sqrt[FullSimplify[Together//@(16DistanceTrilinears[(ExpandAll[TriangleArea[abc]]/.Abs[x_]:>x)^2]),TriangleQ[]]]/4,TriangleQ[]&&0<#<1&/@{α,β,γ}]

(a*b*c*α*β*γ*Sqrt[(-(a - b - c))*(a + b - c)*(a - b + c)*(a + b + c)])/
  Abs[(a*α + b*β - c*γ)*(a*α - b*β + c*γ)*((-a)*α + b*β + c*γ)]

FullSimplify[Sqrt[FullSimplify[Together//@(16DistanceTrilinears[(ExpandAll[TriangleArea[abc]]/.Abs[x_]:>x)^2]),TriangleQ[]]]/4 Δ/TriangleArea[],TriangleQ[]&&0<#<1&/@{α,β,γ}]

Table

t=Triangle[Table[Random[Real,{0,1},30],{3},{2}]];
kc=KimberlingCenters;

Length[k=Vertices[Cartesian[AnticevianTriangle[#][t]]]&/@kc]//Timing

l={};
Do[
Print[l0={
i,
p=Triangles[[i]],
tri=Vertices[p[t]];
kc[[Flatten[Position[k,_?(#==tri&)]]]]//Timing
}];
AppendTo[l,l0],
{i,Length[Triangles]}
]

Nicely Formatted

DisplayForm[Grid[{#[[-1,-1,1]],CenterName[#[[-1,-1,1]]],TriangleName[#[[2]]]}&/@Sort[Select[l,#[[-1,-1]]=!={}&],#1[[-1,-1,1,1]]<#2[[-1,-1,1,1]]&],
RowLines->True,ColumnAlignments->Left
]
]