  We prove that there exists a unique ellipse of minimal eccentricity, E_{I},
inscribed in a parallelogram, D. We also prove that the smallest nonnegative
angle between equal conjugate diameters of E_{I} equals the smallest
nonnegative angle between the diagonals of D. We also prove that if E_{M} is
the unique ellipse inscribed in a rectangle, R, which is tangent at the
midpoints of the sides of R, then E_{M} is the unique ellipse of minimal
eccentricity, maximal area, and maximal arc length inscribed in R. Let D be any
convex quadrilateral. In previous papers, the author proved that there is a
unique ellipse of minimal eccentricity, E_{I}, inscribed in D, and a unique
ellipse, E_{O}, of minimal eccentricity circumscribed about D. We defined D to
be bielliptic if E_{I} and E_{O} have the same eccentricity. In this paper we
show that a parallelogram, D, is bielliptic if and only if the square of the
length of one of the diagonals of D equals twice the square of the length of
one of the sides of D.
