An inellipse 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


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


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.

The centers of the ellipses inscribed in a quadrilateral all lie on the straight line segment joining the midpoints of the polygon diagonals (Chakerian 1979, pp. 136-139).

See also

Brocard Inellipse, Circumellipse, Hofstadter Ellipse, Incircle, Lemoine Inellipse, Mandart Inellipse, MacBeath Inconic, Orthic Inconic, Steiner Inellipse

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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.

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Cite this as:

Weisstein, Eric W. "Inellipse." From MathWorld--A Wolfram Web Resource.

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