Stammler Hyperbola

Author

Eric W. Weisstein
January 12, 2005

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

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

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

Code

Branches[g_]:=Module[{data=Flatten[g][[1,1]]},
split=Split[data,Sqrt[#.#]&[Subtract@@@#]<100&];
{Line/@(Drop[#,-1]&/@split),
Dashing[{.02}],Line[data]
}
]

Equation

ConicJoin[TrilinearVertexMatrix[TangentialTriangle],Trilinears/@{Incenter,SymmedianPoint}]

Simplify[Conic[Join[TrilinearVertexMatrix[TangentialTriangle],Trilinears/@{Incenter,SymmedianPoint}]]]

eqn=(c^2) ((α^2)-(β^2))+(a^2) ((β^2)-(γ^2))+(b^2) (-(α^2)+(γ^2));

Simplify[((ConicCenter[eqn])/((a^2-c^2)(b^2-a^2)(c^2-b^2)))]

FullSimplify[
DistanceTrilinears[TrigExpand[Trilinears[X[110]]]]/(-2Circumradius[]),TriangleQ[]]

TrilinearEquation[StammlerHyperbola]

Figure

t=Triangle[{{0,0},{1,0},{.4,.7}}];
tt=TangentialTriangle[t];
Show[Graphics[{
t,
{Red,
Branches[ParametricPlot[Evaluate[
TrilinearToCartesian[tt,{α,#,1}]&/@(β/.Solve[DistanceTrilinears[eqn]==0/.γ->1/.Thread[{a,b,c}->SideLengths[Evaluate[tt]]],β])],
{α,-100,15},
PlotPoints->150,
DisplayFunction->Identity
][[1]]]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{Incenter[t],"I",Blue},
{SymmedianPoint[t],"K",Blue},
{Circumcenter[t],"O",Blue},
{X[110][t],"(Subscript[X, 110])",Red}
}
],
{PointSize[.025],LabelOffset->.04}
]
}],
PlotRange->{{-.3,1.7},{-.3,1.}},
AspectRatio->Automatic]

Incident Centers

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

First/@ctrs

Length[ctrs]

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