Pedal Triangle

Author

Eric W. Weisstein
December 18, 2004

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

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

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

Figure

t=Triangle[{{0.,0.},{1.,.1},{.5,1.}}];
p=Point[{.4,.4}];
Show[Graphics[{t,
{Red,pt=PedalTriangle[t,p]},
{Blue,lines=PedalLines[t,p],
PerpSquare[#,.05,1]&/@Transpose[{lines,Sides[t]}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[pt],{"A'","B'","C'"}}],
{
{p,"P",Blue}
}
],PointSize[.03],LabelOffset->.04]
}],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

PedalTriangle[{A,B,C},{α,β,γ}]//MatrixForm//TraditionalForm

Side Lengths

abc=Together//@DistanceTrilinears[SideLengths[Evaluate[PedalTriangle[{A,B,C},{α,β,γ}]]]]

abc2=FullSimplify[abc,TriangleQ[]&&0<#<1&/@{α,β,γ}]

FullSimplify[abc2 Circumradius[]/R,TriangleQ[]]

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

Area

area=Simplify[Together//@DistanceTrilinears[Area[Evaluate[PedalTriangle[{A,B,C},{α,β,γ}]]]](Δ/Area[])^3,TriangleQ[]]

Table

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

Length[k=Vertices[Cartesian[PedalTriangle[#][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]}
]

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