Napoleon-Feuerbach Cubic

Author

Eric W. Weisstein
November 18, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Napoleon-FeuerbachCubic.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Red,NapoleonFeuerbachCubic[t]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@{
{Incenter[t],"I"},
{Circumcenter[t],"O"},
{Orthocenter[t],"H"},
{NinePointCenter[t],"Ni"},
{NapoleonPoint[1][t],"N"},
{NapoleonPoint[2][t],"N''"}
},
Append[#,Red]&/@Transpose[{Excenters[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}]
],LabelOffset->.06]
}],AspectRatio->Automatic,PlotRange->{{-.5,1.5},{-1.,1.1}}]//Timing

Trilinear Equation

CenterFunction[NinePointCenter]

TrilinearEquation[NapoleonFeuerbachCubic]

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,NapoleonFeuerbachCubic])//Timing

First/@ctrs

Length[ctrs]

{#,Name[#]}&/@ctrs//TableForm

Derivation

eqn=Plus@@CyclicTrilinears[(a^2(b^2+c^2)-(b^2-c^2)^2)α(c^2β^2-b^2γ^2)/a]

eqn2=Factor[a b c eqn]

IncidentCenters[KimberlingCenters,eqn2]//Timing