Weill Point

Author

Eric W. Weisstein
March 27, 2003

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

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

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

Initialization

<<PlaneGeometry`;

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Violet,ct=ContactTriangle[t]},
{Blue,Incircle[t]},
{Red,Medians[ct]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[ct],{"A'","B'","C'"}}],
{
{WeillPoint[t],"W",Red}
}
],LabelOffset->.04,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

O, I, and W collinear

t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
{Blue,Incircle[t]},
{Red,Circumcircle[t]},
{DarkGreen,LongestLine[Coordinates/@{WeillPoint[t],Circumcenter[t]}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{WeillPoint[t],"W",Purple},
{Incenter[t],"I",Blue},
{Circumcenter[t],"O",Red}
}
],LabelOffset->.04,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

d1=FullSimplify[Together//@Distance[Incenter,WeillPoint],TriangleQ[]]

d2=FullSimplify[DistanceTrilinears[Distance[Circumcenter,Incenter]],TriangleQ[]]

FullSimplify[d1/d2,TriangleQ[]]

FullSimplify[Inradius[]/Circumradius[]/3,TriangleQ[]]