Exterior Angle Bisector

Author

Eric W. Weisstein
April 18, 2004

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.4,.5}}];
Show[Graphics[{
{Purple,Excircles[t]},
{Blue,Sides[t,1]},
{Thickness[.01],t},
{Red,ExteriorAngleBisectors[t,.25]},
{Red,
EqualAngleDivisionTicks[{v[[2]]+.5(v[[2]]-v[[1]]),v[[2]],v[[3]]},2,1,.15,.03,.2],
EqualAngleDivisionTicks[{v[[1]],v[[2]],v[[2]]+.5(v[[2]]-v[[3]])},2,1,.15,.03,.2],
EqualAngleDivisionTicks[{v[[3]]+.5(v[[3]]-v[[1]]),v[[3]],v[[2]]},2,2,.15,.04,.2],
EqualAngleDivisionTicks[{v[[1]],v[[3]],v[[3]]+.5(v[[3]]-v[[2]])},2,2,.15,.03,.2],
EqualAngleDivisionTicks[{v[[1]]-.5(v[[2]]-v[[1]]),v[[1]],v[[3]]},2,3,.15,.03,.2],
EqualAngleDivisionTicks[{v[[2]],v[[1]],v[[1]]+.5(v[[1]]-v[[3]])},2,3,.15,.03,.2]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Purple]&/@Transpose[{VertexPoints[ExcentralTriangle[t]],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}]
],PointSize[.027],LabelOffset->.08]
}],AspectRatio->Automatic,PlotRange->{{-.6,1.6},{-.6,1.1}}]

Angle Bisector Triangles

t=Triangle[{{0,0},{1,0},{.4,.5}}];
Show[Graphics[
{
PointSize[.02],
{Blue,LongestLine[vs]},
{Red,ExteriorAngleBisectorTriangles[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Point/@(vs=(TrilinearToCartesian[t,#]&/@{{0,-1,1},{1,0,-1},{-1,1,0}})),
{"A''","B''","C''"}}
]
],LabelOffset->{0,.15}],
{Thickness[.006],t}
},AspectRatio->Automatic],PlotRange->All]

t=Triangle[{{0,0},{1,0},{.2,.5}}];
v1=EndPoint/@Most[AngleBisectors[t]];
v2=VertexPoints[ExteriorAngleBisectorTriangles[t][[3]]][[1]];
Show[Graphics[
{
PointSize[.02],
{Red,ExteriorAngleBisectorTriangles[t][[3]]},
{Red,Most[AngleBisectors[t]]},
{Thickness[.006],t},
{Blue,Line[Coordinates/@Append[v1,v2]]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{v1,{"A'","B'"}}],
Append[#,Red]&/@{{v2,"C''"}}
],LabelOffset->{0,.07}]
},AspectRatio->Automatic],PlotRange->All]

Last/@Coordinates/@AngleBisectors[t]

TrilinearToCartesian[t,#]&/@{{0,1,1},{1,0,1},{1,1,0}}