Jerabek Antipode

Author

Eric W. Weisstein
November 13, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Blue,NinePointCircle[t],Line[Coordinates/@{X[113][t],X[125][t]}]},
{Red,JerabekHyperbola[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[125][t],"(Subscript[X, 125])",Red}},
{{NinePointCenter[t],"Ni",Blue},{X[113][t],"(Subscript[X, 113])",Blue}}
],PointSize[.025],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->{{-.3,1.2},{-.3,.7}}]

Derivation

Through[{Identity,Name,Trilinears}[#]]&/@X/@{113,125}

soln=Solve[{TrilinearEquation[TrilinearLine[Line[X[125],NinePointCenter]]]==0,TrilinearEquation[NinePointCircle]==0,
2Area==a α+b β+c γ},{α,β,γ}];

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

{X[113][t],X[125][t]}

Cartesian[({α,β,γ}/.soln[[1]])[t]]//Timing

Cartesian[({α,β,γ}/.soln[[2]])[t]]//Timing

(expr=(ToSidelengths[α/.soln[[2]]/.trig:(_Cos|_Sin):>TrigExpand[trig]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]])/.Sqrt[s_]:>Simplify[Sqrt[s],TriangleQ[]])//Timing

exprs=expr/.Sqrt[s_]:>Simplify[Sqrt[Factor[s]],TriangleQ[]]

expr1=Factor/@Factor[Coefficient[Expand[exprs],coef=(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]]

expr2=Factor/@Simplify[Factor[Expand[exprs]-expr1 coef],TriangleQ[]]

Simplify[Subtract@@{exprs,expr1  coef+expr2},TriangleQ[]]

res=FullSimplify[expr1  coef+expr2,TriangleQ[]]

Refine[ToAngles[Factor[Simplify[Factor//@Factor[Simplify[res,Sign[((a^6)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))-(a^2) ((b^4)-3 (b^2) (c^2)+(c^4))) ((a^8)+(a^6) (-3 (b^2)+(c^2))+(c^2) (((-(b^2)+(c^2)))^3)+(a^4) (3 (b^4)+2 (b^2) (c^2)-4 (c^4))+(a^2) (-(b^6)-2 (b^4) (c^2)+2 (b^2) (c^4)+(c^6)))]==1]],TriangleQ[]]]],Area>0]

Cartesian[CyclicTrilinears[%][t]]

Factor[ToAngles[Refine[ToAngles[Factor//@Factor[Simplify[res,Sign[((a^6)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))-(a^2) ((b^4)-3 (b^2) (c^2)+(c^4))) ((a^8)+(a^6) (-3 (b^2)+(c^2))+(c^2) (((-(b^2)+(c^2)))^3)+(a^4) (3 (b^4)+2 (b^2) (c^2)-4 (c^4))+(a^2) (-(b^6)-2 (b^4) (c^2)+2 (b^2) (c^4)+(c^6)))]==-1]]],Area>0]]]

Cartesian[CyclicTrilinears[%][t]]

CenterFunction[X[113]]

X[113][t]