The following solution comes from Daniel from Northumberland.
Point X moves around the circumference of a circle of diameter AB. This happens because the four angles at the centre of the rhombus (where the diagonals cross) are always 90 degrees each, no matter where X may be. The angle AXB is always a right angle except when X is at A or at B. The points C, X and A are on top of each other when the path of X goes through A, and similarly D, X and B coincide when X goes through B.