Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Show that if the rectangle has the proportions of A4 paper (AB=BC ) then the ratio of the areas of the rhombus and the rectangle is 3:4. Show also that, by choosing a suitable rectangle, the ratio of the area of the rhombus to the area of the rectangle can take any value strictly between 1/2 and 1.

What is the situation if we try to do this problem by constructing a rhombus on the shorter sides of the rectangle?