Abstract We investigate the question of whether a simple closed curve in the plane must contain all four vertices of some rhombus having one diagonal collinear with a specified point. This complements previous research on whether there is a rhombus with a diagonal or side parallel to a given line. We obtain a new proof that a simple closed curve contains the vertices of uncountably many rhombi. We also explore conditions guaranteeing that all points in some region are collinear with diagonals of such rhombi.
Stephen E. Wright (Wed,) studied this question.