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Bielliptic Quadrilateral

A convex quadrilateral is bielliptic if its unique inellipse of minimum eccentricity and unique circumellipse of minimum eccentricity have the same eccentricity. If their common eccentricity is tau, where 0<=tau<1, the quadrilateral is said to be of class tau (Horwitz 2010).

This notion generalizes that of a bicentric quadrilateral. Every bicentric quadrilateral is bielliptic of class 0, while there also exist bielliptic quadrilaterals of positive class that are not bicentric, including both a nontrapezoidal example and a trapezoid that is not a parallelogram (Horwitz 2010).

A parallelogram is bielliptic iff there are a diagonal of length d and a side of length s such that d^2=2s^2. In particular, a rectangle is bielliptic iff it is a square (Horwitz 2012).

See also

Bicentric Quadrilateral, Circumellipse, Eccentricity, Inellipse, Parallelogram

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References

Horwitz, A. "Ellipses of Minimal Area and of Minimal Eccentricity Circumscribed about a Convex Quadrilateral." Aust. J. Math. Anal. Appl. 7, Article 8, 1-12, 2010. https://arxiv.org/abs/0707.2092.Horwitz, A. "Ellipses Inscribed in Parallelograms." Aust. J. Math. Anal. Appl. 9, Article 6, 1-12, 2012. https://ajmaa.org/searchroot/files/pdf/v9n1/v9i1p6.pdf.

Cite this as:

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

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