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.