Due to Thales' theorem "C" and the diameter form a right triangle with the line segment "DC" as its altitude, hence "DC" is the side of a square with the area of the rectangle.
Due to Thales' theorem, "C" and the endpoints "A" and "B" of that diameter form a right triangle with the line segment "DC" as its altitude. By the geometric mean theorem "DC" is the side of a square with the area "pq" of the rectangle.