A bielliptic quadrilateral is a convex quadrilateral for which its unique inellipse of minimum eccentricity
and unique circumellipse of minimum eccentricity
have the same eccentricity. If their common eccentricity
is 
, where 
,
the quadrilateral is said to be of class 
(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 
and a side of length 
such that 
.
In particular, a rectangle is bielliptic iff
it is a square (Horwitz 2012).
