N2004 P1 Q3

Referred to an origin \({O}\), the position vectors of four non-collinear points \({A}\), \({B}\), \({C}\) and \({D}\) are \({\mathbf{a}}\), \({\mathbf{b}}\), \({\mathbf{c}}\) and \({\mathbf{d}}\) respectively.

Given that \(\mathbf{a}-\mathbf{b}=\mathbf{d}-\mathbf{c}\), show that \({ABCD}\) is a parallelogram.

Given also that \(\left| \mathbf{a}-\mathbf{c} \right|=\left| \mathbf{b}-\mathbf{d} \right|\), identify the shape of the parallelogram \({ABCD}\), justifying your answer.[6]