Fermat Points

Author

Eric W. Weisstein
February 1, 2003

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

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

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

Initialization

<<PlaneGeometry`;

Graphics

tri={{0.,0.},{1.5,.6},{2.,-.5}};
f1=FermatPoint[1,tri];
f2=FermatPoint[2,tri];
pts={{f1,"(Subscript[F, 1])"},{f2,"(Subscript[F, 2])"}};

Show[GraphicsArray[Table[
Graphics[{
{Thickness[.015],Triangle[tri]},
{Dashing[{.02,.02}],Line/@EquilateralTriangularize[i,tri]},{RGBColor[1,0,0],FermatPointConstruction[i,tri]},
PointLabels[tri,pts[[i]],.04,{.12,-.12}]
},AspectRatio->Automatic],{i,2}]]]

Collinearity Theorems

d=Det[DistanceTrilinears/@{FermatPoint[1],FermatPoint[2],SymmedianPoint}]

CollinearQ[{
FermatPoint[1],
FermatPoint[2],
SymmedianPoint,
Midpoint[{Centroid,Orthocenter}]
}]//Timing

t=Triangle[{{0,0},{1,0},{.2,.30}}];
gh=Line[Coordinates/@{Centroid[t],Orthocenter[t]}];
g1=Show[Graphics[{
t,
{Blue,gh,
EqualSideTicksAboutMidpoint[gh,1,.02]},
{Red,
LongestLine[Coordinates/@{f1=FermatPoint[1][t],f2=FermatPoint[2][t],m=Midpoint[gh],k=SymmedianPoint[t]},.4]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{{f1,"X",Red},
{f2,"X'",Red},
{k,"K",Red},
{Centroid[t],"G",Blue},
{Orthocenter[t],"H",Blue},
{m,"(Subscript[M, GH])",Red}}
],{PointSize[.025],LabelOffset->.035{1.2,-1.2}}]
}],AspectRatio->Automatic,
PlotRange->All]

f12=Line[Coordinates/@{f1,f2}];
g2=Show[Graphics[{
t,
{Green,NinePointCircle[t]},
{Red,
f12,
EqualSideTicksAboutMidpoint[f12,1,.02]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{{f1,"X",Red},
{f2,"X'",Red},
{Midpoint[f12],"(Subscript[M, XX'])",Blue},
{NinePointCenter[t],"N",Green}}
],{PointSize[.025],LabelOffset->.035{1.2,-1.2}}]
}],AspectRatio->Automatic,PlotRange->All]

Show[GraphicsArray[{g1,g2}]]