Excentral-Hexyl Ellipse

Author

Eric W. Weisstein
January 29, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Excentral-HexylEllipse.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{CadmiumYellow,Sides[t,3]},
{Green,Excircles[t]},
{Blue,HexylTriangle[t]},
{SlateBlue,ExcentralTriangle[t]},
{Purple,HexylHexagon[t]},
{Red,ExcentralHexylEllipse[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,SlateBlue]&/@Transpose[{VertexPoints[HexylTriangle[t]],{"A'","B'","C'"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[ExcentralTriangle[t]],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
{
{Circumcenter[t],"O",Red}
}
],PointSize[.015],LabelOffset->.15]
}],AspectRatio->Automatic]

Equation

TrilinearEquation[ExcentralHexylEllipse]//TraditionalForm

Center

-Symmetrize/@ToAngles[Symmetrize/@Factor[Center[ExcentralHexylEllipse]]]

CenterFunction[Circumcenter]

Area

CircumellipseArea[ellipse_]:=Module[{
eqn=TrilinearEquation[ellipse],
lines=TrilinearEquation[ParallelLine[Center[ellipse],TrilinearLine[#]]]&/@IdentityMatrix[3],
solns,abc
},
solns=({α,β,γ}/.Solve[{eqn==0,#==0,(a α+b β+c γ)==2Area},{α,β,γ}])&/@lines;
abc=(Distance@@#)^2&/@solns;
Pi Sqrt[Times@@abc]Area/(2a b c)
]

Together//@CircumellipseArea[ExcentralHexylEllipse]

Refine[Factor//@%8,TriangleQ[]]

CyclicProduct[(a^4)-(a^3) b-(a^2) (b^2)+a (b^3)+(a^2) b c+2 a (b^2) c+(b^3) c-2 (a^2) (c^2)+a b (c^2)-(b^2) (c^2)-b (c^3)+(c^4)]==(((a^4)-2 (a^2) (b^2)+(b^4)-(a^3) c+(a^2) b c+a (b^2) c-(b^3) c-(a^2) (c^2)+2 a b (c^2)-(b^2) (c^2)+a (c^3)+b (c^3)) ((a^3) b-(a^2) (b^2)-a (b^3)+(b^4)+(a^3) c+2 (a^2) b c+a (b^2) c-(a^2) (c^2)+a b (c^2)-2 (b^2) (c^2)-a (c^3)+(c^4)) ((a^4)-(a^3) b-(a^2) (b^2)+a (b^3)+(a^2) b c+2 a (b^2) c+(b^3) c-2 (a^2) (c^2)+a b (c^2)-(b^2) (c^2)-b (c^3)+(c^4)))

Incident Centers

IncidentCenters[KimberlingCenters,eqn]//Timing

CenterName[X[1768]]

Factor[CenterFunction[X[1768]]]

Derivation

Factor[Together//@ToSidelengths[Conic[Take[Join@@(TrilinearVertexMatrix/@{HexylTriangle,ExcentralTriangle}),5]]]]

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

With[{l={(α^2),(β^2),(γ^2),α β,β γ,γ α}},Factor[Coefficient[eqn,#]&/@l].l]

InputForm[%13]

Check

Together//@ToSidelengths[(eqn/.Thread[{α,β,γ}->#])&/@Join@@(TrilinearVertexMatrix/@{HexylTriangle,ExcentralTriangle})]