Rhombi inscribed in simple closed curves

We show that given any simple closed curveJ in ² and any lineL, the curveJ contains the four vertices of some rhombusR with two sides parallel toL. Furthermore, the cyclic order of the vertices ofR agrees with their cyclic order onJ. We also show that the diameters of the rhombi so produced (one for each lineL) may be bounded away from zero.