Antiparallel

Author

Eric W. Weisstein
January 12, 2011

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

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Usage

?Antiparallel

One line antiparallel to side a-c

tri=Triangle[{{0.,0.},{.5,1.},{1.,.1}}];
v=Vertices[tri];
r=.6;
anti=Antiparallel[tri,r];
pts={
v[[1]],A,v[[2]],B,v[[3]],C,
anti[[1,1]],(Subscript[A, A]),anti[[1,2]],(Subscript[A, C])
};

Show[Graphics[{
tri,
PointLabels[tri,pts,.04,{.03,.03}],
anti,
{Dashing[{.02,.02}],AntiparallelCircle[tri,r]}
}],AspectRatio->Automatic]

Relationship to symmedian (Lemoine) point

tri=Triangle[{{0.,0.},{.5,1.},{1.,.1}}];
v=Vertices@tri;
pts={
v[[1]],"A",v[[2]],"B",v[[3]],"C",
SymmedianPoint,"K"
};

Show[Graphics[{
tri,
PointLabels[tri,pts,.04,{.03,.03}],
Antiparallel[tri,.6],Antiparallel[tri,.3],
Antiparallel[tri,.53],Antiparallel[tri,.8],
Antiparallel[tri,.7],Antiparallel[tri,.2],
Dashing[{.02,.02}],Symmedians[tri][[2]]
}],AspectRatio->Automatic,PlotRange->All]

Antiparallels through a point

pt=Point[{.4,.15}];
tri=Triangle[{{0.,0.},{.5,.5},{1.,-.2}}];
redpts={pt,"P"};
v=Vertices@tri;
pts={v[[1]],"A",v[[2]],"B",v[[3]],"C"};

Show[Graphics[{
tri,
PointLabels[tri,pts,.03,.05],
{
Red,
Antiparallels[tri,pt],
PointLabels[tri,redpts,.03,.05]
}
}],AspectRatio->Automatic,PlotRange->All]