Euler Line

Author

Eric W. Weisstein
August 2, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Line Equation

LineEquation[EulerLine]

Symmetrize[Factor[ToSidelengths[LineEquation[EulerLine]]]]

Factor/@(Coefficient[%,#]&/@{α,β,γ})

Complement and Anticomplement

Symmetrize[ToSidelengths[Line[EulerLine][[1,1]]]]

Symmetrize[ToSidelengths[TrilinearComplement[EulerLine][[1,1]]]]

Symmetrize[ToSidelengths[Anticomplement[EulerLine][[1,1]]]]

Figure

t=Triangle[{{0,0},{3,-2/10},{1,5/2}}];
pts={
{Centroid[t],"G"},{Circumcenter[t],"O"},{DeLongchampsPoint[t],"Z"},
{NinePointCenter[t],"N"},{Orthocenter[t],"H"},{TangentialTriangleCircumcenter[t],"(Subscript[O, t])"}};

Show[Graphics[
{
t,
PointLabels[Transpose[{VertexPoints[t],{"A","B","C"}}]],
Red,
PointLabels[pts],
Red,LongestLine[Coordinates/@First/@pts]
}]]

Isotomic Conjugate

Factor[ToSidelengths[TrilinearEquation[EulerLine]/.Thread[{α,β,γ}->IsotomicConjugate[{α,β,γ}]]]]

isotomic=(a^5) b α β-a (b^5) α β-(a^3) b (c^2) α β+a (b^3) (c^2) α β-(a^5) c α γ+(a^3) (b^2) c α γ-a (b^2) (c^3) α γ+a (c^5) α γ-(a^2) (b^3) c β γ+(b^5) c β γ+(a^2) b (c^3) β γ-b (c^5) β γ;

IncidentCenters[KimberlingCenters,isotomic]

Isogonal Conjugate

Factor[ToSidelengths[TrilinearEquation[EulerLine]/.Thread[{α,β,γ}->IsogonalConjugate[{α,β,γ}]]]]

isogonal=-(a^4) c α β+(b^4) c α β+(a^2) (c^3) α β-(b^2) (c^3) α β+(a^4) b α γ-(a^2) (b^3) α γ+(b^3) (c^2) α γ-b (c^4) α γ+(a^3) (b^2) β γ-a (b^4) β γ-(a^3) (c^2) β γ+a (c^4) β γ;

Factor[FullSimplify[ToSidelengths[TrigExpand[TrilinearEquation[JerabekHyperbola]]],TriangleQ[]]]

IncidentCenters[KimberlingCenters,isogonal]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Check

Do[
Print[{i,ctrs[[i]],
TimeConstrained[FullSimplify[Simplify[DistanceTrilinears[TrigExpand[LineEquation[EulerLine]/.Thread[{α,β,γ}->Trilinears[ctrs[[i]]]]]],TriangleQ[]],TriangleQ[]],100]//Timing}],
{i,Length[ctrs]}]

Simplify[DistanceTrilinears[LineEquation[SoddyLine]/.Thread[{α,β,γ}->Trilinears[#]]/.{Sin[x_-y_]:>TrigExpand[Sin[x-y]],
Sin[2x_]:>TrigExpand[2x],
Sec[x_/2]:>Sqrt[2/(1+Cos[x])]
}],TriangleQ[]]&/@{X[853],X[854],X[855],X[856],X[857],X[859],X[860],X[861],X[862],X[864],X[865],X[866],X[867],X[868]}

Distances

{h,g,o,n}={Orthocenter,Centroid,Circumcenter,NinePointCenter};
(Together//@DistanceTrilinears[Distance[#1,#2]/Distance[#3,#4]])&@@@{
{h,g,g,o},
{h,o,g,o}
}

Simplify[(Together//@DistanceTrilinears[Distance[#1,#2]/Distance[#3,#4]/.
{
Cos[x_]:>TrigExpand[Cos[x]],
Sin[x_]:>TrigExpand[Sin[x]]
}
])&@@@
{
{h,o,n,o},
{h,o,g,n}
},
TriangleQ[]]//Timing