The diagram shows a parallelogram \(ABCD\) with \(A\) and \(C\) on the coordinate axes. The equation of \(AB\) is \(y=2x+1\) and the equation of \(BC\) is \(2y+x=12\).

- Show that \(ABCD\) is a rectangle.
- Find the coordinates of \(A\), \(B\), \(C\) and \(D\).
- Find the area of \(ABCD\).