The points \(A(1,2)\) and \(B(-2,1)\) are two vertices of a rectangle \(ABCD\). The diagonal \(CA\) produced passes through the point \((2,9)\). Calculate the coordinates of the vertices \(C\) and \(D\).

The points \(A(1,2)\) and \(B(-2,1)\) are two vertices of a rectangle \(ABCD\). The diagonal \(CA\) produced passes through the point \((2,9)\). Calculate the coordinates of the vertices \(C\) and \(D\).