1. The midpoint of the diagonals is given as \(E \left( 6 \frac{1}{2}, 8 \frac{1}{2} \right)\).

2. Use the midpoint formula for diagonals: \(\frac{x_1 + x_3}{2} = 6.5\) and \(\frac{y_1 + y_3}{2} = 8.5\).

3. Solve the equations for \(A\) and \(C\):

\(y = 3x\) and \(4y = x + 11\).

4. Solving these simultaneously gives \(A(1, 3)\).

5. Use the midpoint formula to find \(C\):

\(x_3 = 2 \times 6.5 - 1 = 12\) and \(y_3 = 2 \times 8.5 - 3 = 14\).

6. Thus, \(C(12, 14)\).

7. Find the equation of \(BC\) using the midpoint and slope:

\(4y = x + 44\).

8. Solve for \(B\) using \(y = 3x\) and \(4y = x + 44\):

\(B(4, 12)\).

9. Use the midpoint formula to find \(D\):

\(x_4 = 2 \times 6.5 - 4 = 9\) and \(y_4 = 2 \times 8.5 - 12 = 5\).

10. Thus, \(D(9, 5)\).