Antipedal Triangle

Author

Eric W. Weisstein
December 18, 2004

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

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

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

Diagram

Cartesian[AntipedalTriangle[SymmedianPoint][t]]

AntipedalLines[t]

t=Triangle[{{0.,0.},{1.,.1},{.5,1.}}];
p=SymmedianPoint;
Show[Graphics[{
t,
 {
Red,
apt=Cartesian[AntipedalTriangle[p][t]]
},
{
Blue,
lines=AntipedalLines[t,p[t]],
PerpSquare[#,.06,2]&/@Transpose[{Sides[apt],lines}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[apt],{"A'","B'","C'"}}],
{
{p[t],"P",Blue}
}
],PointSize[.025],LabelOffset->.06]
}], AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

<<Utilities`Typesetting`

Vertices[AntipedalTriangle[{α,β,γ}]]//TraditionalForm

Side Lengths

abc=FullSimplify[Together//@DistanceTrilinears[SideLengths[Evaluate[AntipedalTriangle[{A,B,C},{α,β,γ}]]]] R/Circumradius[],TriangleQ[]]

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

FullSimplify[LawOfCosines[%],TriangleQ[]]

Area

area=FullSimplify[Together//@DistanceTrilinears[Area[Evaluate[AntipedalTriangle[{A,B,C},{α,β,γ}]]]],TriangleQ[]]

Simplify[area R/Circumradius[]]

Table

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

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

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

{#[[-1,-1,1]],CenterName[#[[-1,-1,1]]],#[[2]]}&/@Sort[Select[l,#[[-1,-1]]=!={}&],#1[[-1,-1,1,1]]<#2[[-1,-1,1,1]]&]//TableForm