  Let g be an arbitrary Jordan loop and let G denote the space of rectangles R
which are inscribed in g in such a way that the cyclic order of the vertices of
R is the same whether it is induced by R or by g. We prove that G contains a
connected set S satisfying one of three properties: 1. S consists of rectangles
of uniformly large area, including a square, and every point of g is the vertex
of a rectangle in S. 2. S consists of rectangles having all possible aspect
ratios, and all but at most 4 points of g are vertices of rectangles in S. 3. S
contains rectangles of every sufficiently small diameter, and all but at most 2
points of g are vertices of rectangles in S.
