As described by Alexander Bogomolny of cut-the-knot.org
, for maximum area of trapezoid, the point of tangency H should be at the midline of AB and DC, thus H is the midpoint of BC.
From the figure:
For the angle theta:
A solution to the same problem has been carried out with the aid of Calculus. You can see how ugly it is without the above analogy of Alexander Bogomolny. See this link to to see the solution by maxima and minima: Solution by Calculus