Feuerbach Antipode

Author

Eric W. Weisstein
November 14, 2005

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FeuerbachAntipode.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[11][t],X[119][t]}]},
{Red,FeuerbachHyperbola[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[11][t],"(Subscript[X, 11])",Red}},
{{NinePointCenter[t],"Ni",Blue},{X[119][t],"(Subscript[X, 119])",Blue}}
],PointSize[.025],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->{{-.3,1.2},{-.3,.7}}]

Derivation

Through[{Identity,Name,Trilinears}[#]]&/@X/@{11,119}

soln=Solve[{TrilinearEquation[TrilinearLine[Line[X[11],NinePointCenter]]]==0,TrilinearEquation[NinePointCircle]==0,
2Area==a α+b β+c γ},{α,β,γ}];

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

{X[11][t],X[119][t]}

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

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

(Sqrt[1-(((((a^2)+(b^2)-(c^2)))^2)/(4 (a^2) (b^2)))]) (Sqrt[1-(((((a^2)-(b^2)+(c^2)))^2)/(4 (a^2) (c^2)))])/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]

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

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//@expr2

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

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

Refine[ToAngles[Factor[Simplify[res,(a^4)-2 (a^3) b-2 (b^4)+2 a b (((b-c))^2)+2 (b^3) c+(b^2) (c^2)-2 b (c^3)+(c^4)+(a^2) ((b^2)+2 b c-2 (c^2))>0]]],Area>0]

Cartesian[CyclicTrilinears[%][t]]

Refine[ToAngles[Factor[Simplify[res,(a^4)-2 (a^3) b-2 (b^4)+2 a b (((b-c))^2)+2 (b^3) c+(b^2) (c^2)-2 b (c^3)+(c^4)+(a^2) ((b^2)+2 b c-2 (c^2))<0]]],Area>0]

Factor[%23]

Symmetrize[CyclicTrilinears[((((a^2) b-(b^3)+(a^2) c-2 a b c+(b^2) c+b (c^2)-(c^3)) ((a^2) b-(b^3)+(a^2) c+(b^2) c+b (c^2)-(c^3)-2 a b c Cos[B-C]))/(16 Area ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3))))]]

Cartesian[CyclicTrilinears[%][t]]

CenterFunction[X[119]]

ToAngles[Factor[Simplify[(ToSidelengths[TrigExpand[CenterFunction[X[119]]]])/.Sqrt[s_]/Sqrt[t_]:>Simplify[Sqrt[s/t],TriangleQ[]],TriangleQ[]]]]

Symmetrize[CyclicTrilinears[%]]

Cartesian[CyclicTrilinears[%][t]]

X[119][t]