Nobbs Points

Author

Eric W. Weisstein
April 23, 2003

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

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

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

Initialization

<<PlaneGeometry`;

Figure

t=Triangle[{{0,0},{1,0},{.35,.6}}];
n=NobbsPoints[t];
ct=ContactTriangle[t];
Show[Graphics[{
{Blue,
Line/@Map[Coordinates,Transpose[{n,VertexPoints[t]}],{2}],
Line/@Map[Coordinates,Transpose[{n,VertexPoints[ct]}],{2}]
},
{DarkGreen,LongestLine[Coordinates/@n]},
t,
{Red,ct},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ct],{"D","E","F"}}],
Append[#,DarkGreen]&/@Transpose[{NobbsPoints[t],{"F'","D'","E'"}}]
]]
}],AspectRatio->Automatic,PlotRange->All
]

Trilinears

csides=Partition[TrilinearVertexMatrix[ContactTriangle],2,1,1];

sides=Partition[IdentityMatrix[3],2,1,1];

a b c(-a+b+c)(a-b+c)(a+b-c) Intersections@@@Transpose[{sides,csides}]

%//TraditionalForm

Check

<<PlaneGeometry`

t=Triangle[{{0,0},{1,0},{.35,.6}}];

NobbsPoints[t]

NobbsPoints[t]

TrilinearToCartesian[t,#]&/@{{-b (a-b+c),a (-a+b+c),0},{0,-(a+b-c) c,b (a-b+c)},{(a+b-c) c,0,-a (-a+b+c)}}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]