Circumnormal Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Equilateral

EquilateralQ[CircumnormalTriangle[RandomTriangle[20]]]

Simplify[#^2&/@EquilateralQ[CircumnormalTriangle]/.C->π-A-B]

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,Circumcircle[t]},
{Red,Dashing[{.02}],CircumtangentialTriangle[t]},
{Red,cn=CircumnormalTriangle[t],
VertexArcs[cn,.1],EqualAngleTicks[cn,1,.1,.03]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[cn],{"A'","B'","C'"}}]
],LabelOffset->.04,PointSize[.04]]
}],AspectRatio->Automatic,ImageSize->200,PlotRange->All]

t=Triangle[{{0,0},{1,0},{0,.5}}];
Show[Graphics[{
t,
{Blue,Circumcircle[t]},
{Red,cn=CircumnormalTriangle[t],
VertexArcs[cn,.1],EqualAngleTicks[cn,1,.1,.03]},
{Green,Line/@Transpose[Vertices/@{t,cn}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[cn],{"A'","B'","C'"}}]
],LabelOffset->.04,PointSize[.04]]
}],AspectRatio->Automatic,ImageSize->200,PlotRange->All]

Area

Area[CircumnormalTriangle]

Centers

(pts={#,KimberlingLookup[#[CircumnormalTriangle]]}&/@Take[KimberlingCenters,10]);//Timing

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm