Fermat Axis

Author

Eric W. Weisstein
February 6, 2005

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

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

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

Initialization

Figure

t=Triangle[{{0,0},{1,0},{.3,.4}}];
Show[Graphics[{
t,
{Red,
LongestLine[Coordinates[FermatPoint[#][t]]&/@{1,2},.5]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@{{FermatPoint[1][t],"X"},{FermatPoint[2][t],"X'"}},
{{SymmedianPoint[t],"K",Blue}}
],LabelOffset->{.05,0}]
}],AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.2,.6}}]

Line Equation

LineEquation[FermatAxis]

Trilinear polar

TrilinearPolar[X[476]]

Symmetrize[Simplify[Together//@DistanceTrilinears[TrigExpand[%]],TriangleQ[]][[1]]]

?TrilinearPole

Line Function

LineFunction[FermatAxis]

IncidentCenters[KimberlingCenters,FermatAxis]//Timing

X(1511) is a Kimberling typo

Symmetrize[Simplify[Together//@DistanceTrilinears[TrigExpand[CenterFunction[X[526]]]],TriangleQ[]]]

CenterName[X[526]]

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,FermatAxis])//Timing

First/@ctrs

Length[ctrs]

{#,CenterName[#]}&/@ctrs//TableForm

Derivation

Simplify[DistanceTrilinears[TrigExpand[LineEquation[Trilinears/@FermatPoint/@{1,2}]]],TriangleQ[]]

eqn=Symmetrize[Numerator[%]]

Factor[Coefficient[eqn,#]&/@{α,β,γ}]