Kiepert Antipode

Author

Eric W. Weisstein
November 14, 2005

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/KiepertAntipode.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[115][t],X[114][t]}]},
{Red,KiepertHyperbola[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[115][t],"(Subscript[X, 115])",Red}},
{{NinePointCenter[t],"Ni",Blue},{X[114][t],"(Subscript[X, 114])",Blue}}
],PointSize[.025],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->{{-.3,1.2},{-.3,.7}}]

Derivation

Through[{Identity,Name,Trilinears}[#]]&/@X/@{114,115}

soln=Solve[{TrilinearEquation[TrilinearLine[Line[X[115],NinePointCenter]]]==0,TrilinearEquation[NinePointCircle]==0,
2Area==a α+b β+c γ},{α,β,γ}];

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

{X[114][t],X[115][t]}

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

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

(expr=ToSidelengths[α/.soln[[2]]/.c_Cos:>TrigExpand[c]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]);//Timing

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

expr2=Simplify[Factor[Expand[expr]-expr1 coef],TriangleQ[]]

expr2s=Factor[Simplify[Factor//@Simplify[Factor//@Factor[Simplify[Factor/@Factor[expr2],TriangleQ[]]],TriangleQ[]],TriangleQ[]]]

Simplify[Subtract@@{expr,expr1  coef+expr2s},TriangleQ[]]

res=Simplify[expr1  coef+expr2s,TriangleQ[]]

Factor[Refine[ToAngles[Factor[Simplify[res,(a^6) (c^2)+(b^4) (c^4)+(a^4) ((b^4)-(b^2) (c^2)-2 (c^4))+(a^2) (-(b^6)+2 (b^4) (c^2)-(b^2) (c^4)+(c^6))<(b^6) (c^2)]]],Area>0]]

Cartesian[CyclicTrilinears[((a ((a^2) (b^2)-(b^4)+(a^2) (c^2)-(c^4)) ((a^2)-b c Cos[B-C]))/(8 Area ((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))))][t]]

Factor[Refine[ToAngles[Factor[Simplify[res,(a^6) (c^2)+(b^4) (c^4)+(a^4) ((b^4)-(b^2) (c^2)-2 (c^4))+(a^2) (-(b^6)+2 (b^4) (c^2)-(b^2) (c^4)+(c^6))>(b^6) (c^2)]]],Area>0]]

Cartesian[CyclicTrilinears[(((((b-c))^2) (((b+c))^2) (Sqrt[(b^2) (c^2)-(b^2) (c^2) Cos[2 A]]))/(2 (Sqrt[2]) a ((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))))][t]]

CenterFunction[X[114]]

ToAngles[Factor[Simplify[(Together//@ToSidelengths[TrigExpand[CenterFunction[X[114]]]])/.Sqrt[s_]:>FullSimplify[Sqrt[s],TriangleQ[]],TriangleQ[]]]]

Cartesian[CyclicTrilinears[%][t]]

X[114][t]