Inellipse

Author

Eric W. Weisstein
December 4, 2005

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

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

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

Code

InellipseArea[ellipse_]:=InellipseArea[Center[ellipse]]
InellipseArea[ctr_List?VectorQ]:=Pi Sqrt[Times@@(1-#&/@(ExactTrilinears[ctr]{a,b,c}/Area))] Area

Examples

General

eqn=Pi Sqrt[Times@@(1-#&/@(ExactTrilinears[{α,β,γ}]{a,b,c}/Area))]Area

area=Area π √((1-((2 a α)/(a α+b β+c γ))) (1-((2 b β)/(a α+b β+c γ))) (1-((2 c γ)/(a α+b β+c γ))))/.{α->c y+b z,β->a z+c x,γ->b x +a y}

Refine[Factor//@area,TriangleQ[]]

Sqrt[(π Sqrt[Area Times@@((Area-#)&/@(ExactTrilinears[{α,β,γ}]{a,b,c}))])^2]

%/.Area->1

Brocard Inellipse

Refine[Factor//@Area[BrocardInellipse],TriangleQ[]]

Hofstadter Ellipse E(1/2)

Refine[Together//@InellipseArea[CyclicTrilinears[b+c]],TriangleQ[]]

Incircle

Refine[Factor//@Refine[ToSidelengths[InellipseArea[Incircle]],TriangleQ[]],TriangleQ[]]

ToSidelengths[π (Inradius^2)]

%((Area^2)/(Area^2//ToSidelengths))

Lemoine Inellipse

Factor//@Refine[Area[LemoineInellipse],TriangleQ[]]

Mandart Inellipse

Refine[Factor//@Area[MandartInellipse],TriangleQ[]]

%((Area^2)/(Area^2//ToSidelengths))

MacBeath Inconic

Refine[ToAngles[Refine[Factor//@ToSidelengths[Factor//@Area[MacBeathInconic]],TriangleQ[]]],TriangleQ[]]

Orthic Inconic

Refine[ToAngles[Refine[Together//@ToSidelengths[Area[OrthicInconic]],TriangleQ[]]],Area>0&&TriangleQ[]]

Steiner Inellipse

Refine[Area[SteinerInellipse],TriangleQ[]]