Triangulation Point

Author

Eric W. Weisstein
February 6, 2003

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

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

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

Initialization

<<PlaneGeometry`

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
pt=Point[{.4,.2}];
Show[Graphics[{
{Blue,TriangulationTriangles[t,pt]},
t,
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{pt,"P",Blue}
}
],
{PointSize[.025],LabelOffset->.06}
]
}],AspectRatio->Automatic,PlotRange->All]
Null

Triangulation Triangle Theorem

TriangulationTheorem[t_Triangle,{pt_,plabel_},{ctr_,clabel_}]:=Module[{
v=Point/@Vertices[t],
l=pt[t]
},
lt=Triangle[{Coordinates[l],Sequence@@#}]&/@Partition[RotateLeft[Vertices[t]],2,1,1];
ls=pt/@lt;
Graphics[{
PointLabels[Transpose[{v,{"A","B","C"}}]],
{Red,
lt,
PointLabels[{{l,plabel}}]
},
{Blue,
Line/@Transpose[{Vertices[t],Coordinates/@ls}],
PointLabels[Transpose[{ls,Subscript[plabel,#]&/@{"A","B","C"}}]]
},
PointLabels[{{ctr[t],clabel,DarkGreen}}],
t
},AspectRatio->Automatic,PlotRange->All]
]

t=Triangle[{{0.,0.},{1,0},{.3,.7}}];
Show[GraphicsArray[Partition[
{
TriangulationTheorem[t,{Circumcenter,"O"},{X[54],"Ko"}],
TriangulationTheorem[t,{FermatPoint[1],"X"},{FermatPoint[1],"X"}],
TriangulationTheorem[t,{Incenter,"I"},{X[1127],"(Subscript[X, 1127])"}],
TriangulationTheorem[t,{SymmedianPoint,"K"},{X[251],"(Subscript[X, 251])"}],
TriangulationTheorem[t,{Centroid,"G"},{Centroid,"G"}]
},
3,3,{1,1},{}]
]]