I was sitting down, solving one of Catriona Shearer‘s geometry puzzles when I found this (by myself):

If we call the long diagonal x, and the line segment perpendicular to the diagonal y, then the area of the triangle = xy/2, and the rectangle is made up of two of those triangles with area xy/2, so the total area of the rectangle = (xy/2) * 2 = xy.

I guess there could be another way to prove it. But this is the neat way I found!