Circumellispe

Author

Eric W. Weisstein
December 4, 2005

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

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

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

Code

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)
]

Examples

General

eqn=(x/α)+(y/β)+(z/γ);

lines=TrilinearEquation[ParallelLine[{x(-a x+b y+c z),y(a x-b y+c z),z(a x+b y-c z)},TrilinearLine[#]]]&/@IdentityMatrix[3]

solns=Factor//@Refine[Factor//@(({α,β,γ}/.Solve[{eqn==0,#==0,(a α+b β+c γ)==2Area},{α,β,γ}])&/@lines),Area>0]

abc=(Distance@@#)^2&/@solns;

Refine[Sqrt[Together//@(Times@@abc)],x>0&&y>0&&z>0&&TriangleQ[]]

Pi %Area/(2a b c)

((4 a b c π x y z )/(((2 (a b x y+a c x z+b c y z)-((a^2) (x^2)+(b^2) (y^2)+(c^2) (z^2))))^(3/2)))Area

Circumcircle

Refine[Together//@ToSidelengths[CircumellipseArea[Circumcircle]],TriangleQ[]&&Area>0]

ToAngles[Factor//@%6]

ToAngles[ToSidelengths[π Circumradius^2]]

Apart/@Apart[((TrilinearEquation[Circumcircle])/(α β γ))]

Refine[ToAngles[Factor//@(((4 a b c π x y z )/(((2 (a b x y+a c x z+b c y z)-((a^2) (x^2)+(b^2) (y^2)+(c^2) (z^2))))^(3/2)))Area/.Thread[{x,y,z}->CyclicTrilinears[a]])],Area>0]

Excentral-Hexyl Ellipse

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)))

Hofstadter Ellipe E(0)

TrilinearEquation[HofstadterEllipse[0]]:=((β γ Sin[A])/A)+((α γ Sin[B])/B)+((α β Sin[C])/C)
Unprotect[Center];
Center[HofstadterEllipse[0]]:={a/(A(B-C)),b/(B(C-A)),c/(C(A-B))}

area=Together//@CircumellipseArea[HofstadterEllipse[0]]

Refine[Sqrt[Factor[area^2/.f:(_Sin|_Cos|_Csc):>ToSidelengths[f]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]]],Area>0&&TriangleQ[]]

%13/.Abs[x_]:>x

MacBeath Circumconic

Refine[ToAngles[Map[Factor,Refine[Factor//@ToSidelengths[Area[MacBeathCircumconic]],TriangleQ[]],{2}]],Area>0&&TriangleQ[]]

Steiner Circumellipse

Area[SteinerCircumellipse]