Feuerbach Hyperbola

Author

Eric W. Weisstein
January 29, 2005

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

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

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

Equation

Hyperbola Parameters

{x,y,z}={Cos[B]-Cos[C],Cos[C]-Cos[A],Cos[A]-Cos[B]};

Hyperbola Condition

Simplify[TrigExpand[DistanceTrilinears[x Cos[A]+y Cos[B]+z Cos[C]]],a>0&&b>0&&c>0]

Equation

eqn=x/α+y/β+z/γ

Figure

t=Triangle[{{0,0},{1,0},{.4,.7}}];
Show[Graphics[{
t,
{Red,FeuerbachHyperbola[t]},
{Purple,NinePointCircle[t]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{{FeuerbachPoint[t],"F",Red}},
Append[#,Blue]&/@{
{Orthocenter[t],"H",LabelOffset->{-.02,.05}},
{Incenter[t],"I",LabelOffset->{.04,.02}},
{GergonnePoint[t],"Ge",LabelOffset->-.04},
{NagelPoint[t],"Na"},
{Mittenpunkt[t],"M",LabelOffset->-.04},
{SchifflerPoint[t],"Sc"}
}
],
{PointSize[.018],LabelOffset->.04}
]
}],
PlotRange->{{-.4,1.5},{-.4,1}},
AspectRatio->Automatic,ImageSize->350]

CenterName[X[#]]&/@{1,4,7,8,9,21,79}

CenterName[X[11]]

Incident Centers

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

old={X[1],X[4],X[7],X[8],X[9],X[21],X[79],X[80],X[84],X[90],X[104],X[177],X[256],X[294],X[314],X[885],X[941],X[943],X[981],X[983],X[987],X[989],X[1000],X[1039],X[1041],X[1061],X[1063],X[1156],X[1172],X[1251],X[1320],X[1389],X[1392],X[1476],X[1896],X[1937],X[2298],X[2320],X[2335],X[2344],X[2346],X[2481],X[2648],X[2997]};

Complement@@@{{old,ctrs},{ctrs,old}}

First/@ctrs

Length[ctrs]

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

Check

Simplify[DistanceTrilinears[TrigExpand[eqn/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs//Timing