Jerabek Hyperbola

Author

Eric W. Weisstein
January 4, 2005

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

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

Hyperbola Parameters

{x,y,z}={a(Sin[2B]-Sin[2C]),b(Sin[2C]-Sin[2A]),c(Sin[2A]-Sin[2B])};

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

TrilinearEquation[JerabekHyperbola]

FullSimplify[ToSidelengths/@((a((b^2)-(c^2))SA β γ+b((c^2)-(a^2))SB γ α+c((a^2)-(b^2))SC α β)/(TrigExpand[TrilinearEquation[JerabekHyperbola]])),TriangleQ[]]

Figure

t=Triangle[{{0,0},{1,0},{.4,.7}}];
Show[Graphics[{
t,
{Red,
Branches[ParametricPlot[Evaluate[
TrilinearToCartesian[t,{α,#,1}]&/@(β/.Solve[DistanceTrilinears[eqn]==0/.γ->1/.Thread[{a,b,c}->SideLengths[t]],β])],
{α,-20,20},
PlotPoints->100,
DisplayFunction->Identity
][[1]]]
},
{Purple,NinePointCircle[t]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{{X[125][t],"(Subscript[X, 125])",Red}},
Append[#,Blue]&/@{
{Orthocenter[t],"H"},
{Circumcenter[t],"O"},
{SymmedianPoint[t],"K"},
{IsogonalConjugate[t,NinePointCenter[t]],"N'"},
{IsogonalConjugate[t,DeLongchampsPoint[t]],"L'"}
}
],
{PointSize[.025],LabelOffset->-.08}
]
}],
PlotRange->{{-.5,1.5},{-.5,1}},
AspectRatio->Automatic]

CenterName[X[#]]&/@{3,4,6,54,64,65,66,67,68}

Incident Centers

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

old={X[3],X[4],X[6],X[54],X[64],X[65],X[66],X[67],X[68],X[69],X[70],X[71],X[72],X[73],X[74],X[248],X[265],X[290],X[695],X[879],X[895],X[1173],X[1175],X[1176],X[1177],X[1242],X[1243],X[1244],X[1245],X[1246],X[1439],X[1798],X[1903],X[1942],X[1987],X[2213],X[2435],X[2574],X[2575],X[2992],X[2993]};

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

First/@ctrs

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

<<MathWorld

{#,CenterName[X[#]]}&/@{20,22,23,26,27,28,237,384,468}//TableForm

Check

Simplify[DistanceTrilinears[TrigExpand[eqn/.Thread[{α,β,γ}->Trilinears[X[#]]]]],
TriangleQ[]]&/@x[[{-1}]]//Timing

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

FullSimplify[{((a^2)-(b^2)) ((a^2)-(c^2)) ((b^2)-(c^2)) ,((((a^3)-a (((b-c))^2)+(a^2) (b+c)-(((b-c))^2) (b+c)))^2) ,((a^3)-(b^3)-(b^2) c-b (c^2)-(c^3)-(a^2) (b+c)+a ((b^2)+(c^2))) ,(√(-(((a-b-c))^3) ((a^3)-a (((b-c))^2)+(a^2) (b+c)-(((b-c))^2) (b+c)))+a (Sqrt[-(a^4)-((((b^2)-(c^2)))^2)+2 (a^2) ((b^2)+(c^2))])-b (Sqrt[-(a^4)-((((b^2)-(c^2)))^2)+2 (a^2) ((b^2)+(c^2))])-c (Sqrt[-(a^4)-((((b^2)-(c^2)))^2)+2 (a^2) ((b^2)+(c^2))]))},TriangleQ[]]

Mathematica can't do the above simplification, but it is true