The deckchair principle.

part (a)

As the rectangle collapses into a parallelogram, like a folding deckchair, 'something' happens to one side of the new quarilateral (i.e. the square). Let's assume wlog it gets shorter. 

But that side is the distance between the centre of a large square and the centre of a small square, so whatever happened to that side must by symmetry also have happened to the other 3 identical sides which are also the distances between the centre of a large square and the centre of a small square.

So the quadrilateral remains a square.

Similarly, when the parallelogram is a rectangle, the required diagonals are trivially concurrent. As the rectangle collapses into a parallelogram, it remains a square as shown above,  so its diagonals continue to meet at the same point as those of the parallelogram.