Yff Hyperbola

Author

Eric W. Weisstein
January 16, 2005

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

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

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

Figure

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

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Red,
Branches[ParametricPlot[Evaluate[TrilinearToCartesian[t,
{Cos[A]+Sin[B-C+u],Cos[B]+Sin[C-A+u],Cos[C]+Sin[A-B+u]}/.Thread[{A,B,C}->Angles[t]]]],{u,-4,4},
PlotPoints->100,
DisplayFunction->Identity
][[1]]]
},
{Blue,
EulerLine[t,2],
l1=PerpendicularLine[NinePointCenter[t],l2=Line[Coordinates/@{Circumcenter[t],Orthocenter[t]}],1],
PerpSquare[{l1,l2},.05,4]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
{
{Centroid[t],"G",Red},
{Orthocenter[t],"H",Red},
{Circumcenter[t],"O",Blue},
{NinePointCenter[t],"N",Blue}
}
],
{PointSize[.03],LabelOffset->.07}
]
}],AspectRatio->Automatic,
PlotRange->{{-.5,1.2},{-.5,1.1}}
]

Equation

TrilinearEquation[YffHyperbola]//TraditionalForm

Factor[Coefficient[TrilinearEquation[YffHyperbola],#]&/@{(α^2),(β^2),(γ^2),α β,β γ,γ α}]

Factor[ToAngles[%]]

Center

ctr=Symmetrize/@Center[YffHyperbola]

IncidentCenters[{X[381]},ctr]//Timing

CenterName[X[381]]

Distances

Semimajor Axis

(a1=Factor//@Simplify[Factor[Factor[Together//@ToSidelengths[Distance[X[381],Centroid]/.f:_Cos:>TrigExpand[f]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]](Circumradius/(ToSidelengths[Circumradius])),TriangleQ[]])//TraditionalForm

(a2=Factor//@Simplify[Factor[Factor[Together//@ToSidelengths[Distance[X[381],Orthocenter]/.f:_Cos:>TrigExpand[f]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]](Circumradius/(ToSidelengths[Circumradius])),TriangleQ[]])//TraditionalForm

a1==a2

Focal Length

(cc=Factor//@Simplify[Factor[Factor[Together//@ToSidelengths[Distance[X[381],Circumcenter]/.f:_Cos:>TrigExpand[f]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]](Circumradius/(ToSidelengths[Circumradius])),TriangleQ[]])//TraditionalForm

cc/a1

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Derivation

Center[eqn]

Symmetrize/@Factor[ToSidelengths[Midpoint[{Centroid,Orthocenter}]]]

Factor[ToAngles[%9]]

IncidentCenters[KimberlingCenters,%9]//Timing

CenterName[X[381]]

CenterFunction[X[381]]

Solve[eqns,{u,v,w,f,g,h}]

Trilinears[Circumcenter]

Circumradius//ToSidelengths

ToAngles[FullSimplify[ToSidelengths[ExactTrilinears[Circumcenter]],TriangleQ[]]]

?Reflection

op0=Factor[Simplify[ToSidelengths[Reflection[Circumcenter,X[381]]/.f_Cos:>TrigExpand[f]],TriangleQ[]]]

op=Factor//@Simplify[Factor[ToSidelengths[ExactTrilinears[op0]]],TriangleQ[]]

k=Together//@ToSidelengths[Distance[Centroid,Orthocenter]]

r1=ExactDistance[{α,β,γ},op];
r2=ExactDistance[{α,β,γ},Circumradius Trilinears[Circumcenter]];

{r1,r2,d}

Refine[Factor[Together//@ToSidelengths[r1]],TriangleQ[]]

r1=(√(7 (a^10)-21 (a^8) (b^2)+14 (a^6) (b^4)+14 (a^4) (b^6)-21 (a^2) (b^8)+7 (b^10)-21 (a^8) (c^2)+26 (a^6) (b^2) (c^2)-10 (a^4) (b^4) (c^2)+26 (a^2) (b^6) (c^2)-21 (b^8) (c^2)+14 (a^6) (c^4)-10 (a^4) (b^2) (c^4)-10 (a^2) (b^4) (c^4)+14 (b^6) (c^4)+14 (a^4) (c^6)+26 (a^2) (b^2) (c^6)+14 (b^4) (c^6)-21 (a^2) (c^8)-21 (b^2) (c^8)+7 (c^10)+60 (a^7) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-72 (a^5) (b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-36 (a^3) (b^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α+48 a (b^6) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-72 (a^5) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α+120 (a^3) (b^2) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-48 a (b^4) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-36 (a^3) (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-48 a (b^2) (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α+48 a (c^6) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α+36 (a^8) (α^2)-108 (a^6) (b^2) (α^2)+108 (a^4) (b^4) (α^2)-36 (a^2) (b^6) (α^2)-108 (a^6) (c^2) (α^2)+72 (a^4) (b^2) (c^2) (α^2)+36 (a^2) (b^4) (c^2) (α^2)+108 (a^4) (c^4) (α^2)+36 (a^2) (b^2) (c^4) (α^2)-36 (a^2) (c^6) (α^2)+48 (a^6) b (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-36 (a^4) (b^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-72 (a^2) (b^5) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β+60 (b^7) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-48 (a^4) b (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β+120 (a^2) (b^3) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-72 (b^5) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-48 (a^2) b (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-36 (b^3) (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β+48 b (c^6) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-36 (a^6) (b^2) (β^2)+108 (a^4) (b^4) (β^2)-108 (a^2) (b^6) (β^2)+36 (b^8) (β^2)+36 (a^4) (b^2) (c^2) (β^2)+72 (a^2) (b^4) (c^2) (β^2)-108 (b^6) (c^2) (β^2)+36 (a^2) (b^2) (c^4) (β^2)+108 (b^4) (c^4) (β^2)-36 (b^2) (c^6) (β^2)+48 (a^6) c (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-48 (a^4) (b^2) c (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-48 (a^2) (b^4) c (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ+48 (b^6) c (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-36 (a^4) (c^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ+120 (a^2) (b^2) (c^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-36 (b^4) (c^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-72 (a^2) (c^5) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-72 (b^2) (c^5) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ+60 (c^7) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-36 (a^6) (c^2) (γ^2)+36 (a^4) (b^2) (c^2) (γ^2)+36 (a^2) (b^4) (c^2) (γ^2)-36 (b^6) (c^2) (γ^2)+108 (a^4) (c^4) (γ^2)+72 (a^2) (b^2) (c^4) (γ^2)+108 (b^4) (c^4) (γ^2)-108 (a^2) (c^6) (γ^2)-108 (b^2) (c^6) (γ^2)+36 (c^8) (γ^2)))/(3 (Sqrt[2]) Refine[√Factor[-(a-b-c) (a+b-c) (a-b+c) (a+b+c) (-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4))],TriangleQ[]])

Refine[Factor[Together//@ToSidelengths[r2]],TriangleQ[]]

r2=(√(-(a^10)+3 (a^8) (b^2)-2 (a^6) (b^4)-2 (a^4) (b^6)+3 (a^2) (b^8)-(b^10)+3 (a^8) (c^2)-6 (a^6) (b^2) (c^2)+6 (a^4) (b^4) (c^2)-6 (a^2) (b^6) (c^2)+3 (b^8) (c^2)-2 (a^6) (c^4)+6 (a^4) (b^2) (c^4)+6 (a^2) (b^4) (c^4)-2 (b^6) (c^4)-2 (a^4) (c^6)-6 (a^2) (b^2) (c^6)-2 (b^4) (c^6)+3 (a^2) (c^8)+3 (b^2) (c^8)-(c^10)-4 (a^7) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α+8 (a^5) (b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-4 (a^3) (b^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α+8 (a^5) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-8 (a^3) (b^2) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α-4 (a^3) (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) α+4 (a^8) (α^2)-12 (a^6) (b^2) (α^2)+12 (a^4) (b^4) (α^2)-4 (a^2) (b^6) (α^2)-12 (a^6) (c^2) (α^2)+8 (a^4) (b^2) (c^2) (α^2)+4 (a^2) (b^4) (c^2) (α^2)+12 (a^4) (c^4) (α^2)+4 (a^2) (b^2) (c^4) (α^2)-4 (a^2) (c^6) (α^2)-4 (a^4) (b^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β+8 (a^2) (b^5) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-4 (b^7) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-8 (a^2) (b^3) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β+8 (b^5) (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-4 (b^3) (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) β-4 (a^6) (b^2) (β^2)+12 (a^4) (b^4) (β^2)-12 (a^2) (b^6) (β^2)+4 (b^8) (β^2)+4 (a^4) (b^2) (c^2) (β^2)+8 (a^2) (b^4) (c^2) (β^2)-12 (b^6) (c^2) (β^2)+4 (a^2) (b^2) (c^4) (β^2)+12 (b^4) (c^4) (β^2)-4 (b^2) (c^6) (β^2)-4 (a^4) (c^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-8 (a^2) (b^2) (c^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-4 (b^4) (c^3) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ+8 (a^2) (c^5) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ+8 (b^2) (c^5) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-4 (c^7) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) γ-4 (a^6) (c^2) (γ^2)+4 (a^4) (b^2) (c^2) (γ^2)+4 (a^2) (b^4) (c^2) (γ^2)-4 (b^6) (c^2) (γ^2)+12 (a^4) (c^4) (γ^2)+8 (a^2) (b^2) (c^4) (γ^2)+12 (b^4) (c^4) (γ^2)-12 (a^2) (c^6) (γ^2)-12 (b^2) (c^6) (γ^2)+4 (c^8) (γ^2)))/((Sqrt[2]) Refine[√Factor[-(a-b-c) (a+b-c) (a-b+c) (a+b+c) (-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4))],TriangleQ[]])

r1r2=r1 r2/.Sqrt[x_]Sqrt[y_]:>Sqrt[x y]

eqn=ToSidelengths[(r2-r1)^2-k^2];

t=Triangle[{{0,0},{1,0},{.3,.5}}];

(ToSidelengths[eqn/.Thread[{α,β,γ}->ExactTrilinears[#]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]])&/@{Orthocenter,Centroid}

eqn2=4(r1r2)^2-(d^2-r1^2-r2^2)^2

(ToSidelengths[eqn2/.Thread[{α,β,γ}->ExactTrilinears[#]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]])&/@{Orthocenter,Centroid}

eqn3=Numerator[-Factor[eqn2]/8]

f=Factor[Coefficient[eqn3,(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]]

g=Factor[eqn3-f(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]

eqn4=g+4Area f

(ToSidelengths[eqn4/.Thread[{α,β,γ}->ExactTrilinears[#]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]])&/@{Orthocenter,Centroid}

Factor[ToSidelengths[eqn4/.Thread[{α,β,γ}->ExactTrilinears[{α,β,γ}]]]]

eqn5=((a^10) (α^2)-2 (a^8) (b^2) (α^2)+(a^6) (b^4) (α^2)-2 (a^8) (c^2) (α^2)+3 (a^6) (b^2) (c^2) (α^2)-(a^4) (b^4) (c^2) (α^2)+(a^6) (c^4) (α^2)-(a^4) (b^2) (c^4) (α^2)+2 (a^7) (b^3) α β-4 (a^5) (b^5) α β+2 (a^3) (b^7) α β-(a^7) b (c^2) α β+(a^5) (b^3) (c^2) α β+(a^3) (b^5) (c^2) α β-a (b^7) (c^2) α β+(a^5) b (c^4) α β-3 (a^3) (b^3) (c^4) α β+a (b^5) (c^4) α β+(a^3) b (c^6) α β+a (b^3) (c^6) α β-a b (c^8) α β+(a^4) (b^6) (β^2)-2 (a^2) (b^8) (β^2)+(b^10) (β^2)-(a^4) (b^4) (c^2) (β^2)+3 (a^2) (b^6) (c^2) (β^2)-2 (b^8) (c^2) (β^2)-(a^2) (b^4) (c^4) (β^2)+(b^6) (c^4) (β^2)-(a^7) (b^2) c α γ+(a^5) (b^4) c α γ+(a^3) (b^6) c α γ-a (b^8) c α γ+2 (a^7) (c^3) α γ+(a^5) (b^2) (c^3) α γ-3 (a^3) (b^4) (c^3) α γ+a (b^6) (c^3) α γ-4 (a^5) (c^5) α γ+(a^3) (b^2) (c^5) α γ+a (b^4) (c^5) α γ+2 (a^3) (c^7) α γ-a (b^2) (c^7) α γ-(a^8) b c β γ+(a^6) (b^3) c β γ+(a^4) (b^5) c β γ-(a^2) (b^7) c β γ+(a^6) b (c^3) β γ-3 (a^4) (b^3) (c^3) β γ+(a^2) (b^5) (c^3) β γ+2 (b^7) (c^3) β γ+(a^4) b (c^5) β γ+(a^2) (b^3) (c^5) β γ-4 (b^5) (c^5) β γ-(a^2) b (c^7) β γ+2 (b^3) (c^7) β γ-(a^4) (b^2) (c^4) (γ^2)-(a^2) (b^4) (c^4) (γ^2)+(a^4) (c^6) (γ^2)+3 (a^2) (b^2) (c^6) (γ^2)+(b^4) (c^6) (γ^2)-2 (a^2) (c^8) (γ^2)-2 (b^2) (c^8) (γ^2)+(c^10) (γ^2));

(ToSidelengths[eqn5/.Thread[{α,β,γ}->Trilinears[#]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]])&/@{Orthocenter,Centroid}

Symmetrize/@Center[eqn5]

Factor[Simplify[ToSidelengths[Trilinears[X[381]]/.f:_Cos:>TrigExpand[f]],TriangleQ[]]]

InputForm[eqn5]