This simply stems from the definition of the cross product.
This is somewhat more interesting, it is left as an exercise to the reader that the quad formed by connecting the 4 midpoints of the edges is a parallelogram and further that the area of the quad is half the area of this parallelogram. For more information see Pierre Varignon who is credited with discovering this around 1730.
This general case is somewhat more difficult to derive. One approach is Stokes theorem, another is to decompose the polygon into triangles of quads. In the following N is the normal to the plane on which the polygon lies.
