Let ABHF be a trapezoid such that 1) sides AB and FH are parallel, 2) angle BAF is 90 degrees, and 3) side FH is larger than side AB. Let G be a point on side FH such that line segment BG is perpendicular to FH. Let line segment CDE be parallel to AB with points C, D, and E on line segments AF, BG, and BH respectively. Let x = [ABDC], y = [CDGF], z = [BDE], and w = [DEHG]. Determine w in terms of x, y, and z. Note: [PQ...UV] denotes the area of polygon PQ...UV.