2001 AMC 12 Problem 20

Points A = (3,9), B = (1,1), C = (5,3), and D=(a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD. The quadrilateral formed by joining the midpoints of \overline{AB}, \overline{BC}, \overline{CD}, and \overline{DA} is a square. What is the sum of the coordinates of point D?

\text{(A) }7 \qquad \text{(B) }9 \qquad \text{(C) }10 \qquad \text{(D) }12 \qquad \text{(E) }16

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.