What properties will a minimum enclosing trapezoid over a set of points have?

I want to write an algorithm that find the trapezoid that encloses a set of points (the set is already a convex hull), such that the area of it is minimal.

I've been googling and doodling and I have the impression that at least one side of it is going to be a straight line that is a superset to one of the enclosed polygon's edges.

I'm also almost positive that each of the other three sides will be a superset to an edge of the polygon (like the first one is) or that its middle point will be right at a vertex of the polygon.

Can I make further assumptions? E.g. that there will be two sides that are supersets? Anything else that may be used to additional lower the complexity of the problem?

I have a working algorithm for triangles. Can I reuse it somehow? On one hand, the trapezoid is just a bunch of triangles, so I feel that I can. On the other hand, some portion of such triangles are inside and some other portion is outside the enclosed polygon.

Your second statement is false. It is easy to contrive a set of four points such that there is no trapezoid where all four sides are supersents to one of the four edges defined by the convex hull. For example, $(0,0), (0,100), (1,100), (2,-1)$.
The problem of finding the minimal area trapezoid can be solved in in polynomial time (in fact, in $O(n)$) in the number of points in the convex hull if you first order the points such that the angle from some chosen "first" point, to some selected interior point, to each of the other points is a monotone function of the point number. Once you have that, you can fix, in turn, one of the parallel sides to go through a consecutive pair of points. The minimal trapezoid under that constraint is easy to find; and there are $n$ such choices to evaluate.