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Inellipse

An inellipse is an inconic that is an ellipse.

The locus of the centers of the ellipses inscribed in a triangle is the interior of the medial triangle. Newton gave the solution to inscribing an ellipse in a convex quadrilateral (Dörrie 1965, p. 217).

The area of an inellipse with center having areal coordinates (t,u,v) inscribed in a triangle is

A=pisqrt((1-2t)(1-2u)(1-2v))Delta,
(1)

where Delta is the area of the reference triangle (Chakerian 1979, pp. 143 and 148), which corresponds to an inellipse with center having exact trilinear coordinates alpha:beta:gamma having area

A = [画像:pisqrt((1-(aalpha)/Delta)(1-(bbeta)/Delta)(1-(cgamma)/Delta))Delta]
(2)
= pisqrt(Delta(Delta-aalpha)(Delta-bbeta)(Delta-cgamma)).
(3)

In terms of the inconic parameters x:y:z, the formula is even simpler,

(E. W. Weisstein, Dec. 4, 2005).

The following table summarizes the areas of some special inellipses.

inellipse center area
Brocard inellipse X_(39) (pia^2b^2c^2)/((a^2b^2+b^2c^2+c^2a^2)^(3/2))Delta
Hofstadter ellipse with r=1/2 X_(37) (piabc)/((ab+bc+ca)^(3/2))Delta
incircle X_1 (4piDelta^2)/((a+b+c)^2)
Lemoine inellipse X_(597) [画像:(pisqrt((2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)))/(3sqrt(3)(a^2+b^2+c^2)^(3/2))Delta]
Macbeath inconic X_5 (a^2b^2c^2pisqrt(cosAcosBcosC))/(16sqrt(2)Delta^2)
Mandart inellipse X_9 (16piDelta^3)/((a+b+c)[2ab+2bc+2ca-(a^2+b^2+c^2)]^(3/2))
orthic inconic X_6 (2pisqrt(2)abcsqrt(cosAcosBcosC))/((a^2+b^2+c^2)^(3/2))Delta
Steiner inellipse X_2 pi/(3sqrt(3))Delta

For a convex quadrilateral that is not a parallelogram, the centers of its inellipses are precisely the points of the open line segment joining the midpoints of the polygon diagonals. Each point of this segment is the center of a unique inellipse. For a parallelogram, every inellipse is centered at the common midpoint of the diagonals (Chakerian 1979, pp. 136-139; Horwitz 2005).

Every convex quadrilateral has a unique inellipse of maximal area, namely its John ellipsoid in two dimensions. If E is any ellipse inscribed in a convex quadrilateral Q, then

with equality iff Q is a parallelogram and E is tangent to its sides at their midpoints (Horwitz 2010).

See also

Brocard Inellipse, Circumellipse, Hofstadter Ellipse, Incircle, John Ellipsoid, Lemoine Inellipse, Macbeath Inconic, Mandart Inellipse, Orthic Inconic, Steiner Inellipse

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References

Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., 1979.Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965.Horwitz, A. "Ellipses of Maximal Area and of Minimal Eccentricity Inscribed in a Convex Quadrilateral." Aust. J. Math. Anal. Appl. 2, 1-12, 2005.Horwitz, A. "An Area Inequality for Ellipses Inscribed in Quadrilaterals." J. Math. Inequal. 4, 431-443, 2010. https://doi.org/10.7153/jmi-04-40.

Referenced on Wolfram|Alpha

Inellipse

Cite this as:

Weisstein, Eric W. "Inellipse." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Inellipse.html

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