Could I get a hint on how to go about this problem

Let L1 and L2 be skew lines; let P1, and P2 be any two distinct points on L1 and let Q1 and Q2 be any two distinct points on L2; let M be a plane parallel to both L1 and L2. Show that the intersections of M with the four lines determined by the line-segments P1Q1, P1Q2,P2Q1, and P2Q2 form the vertices of a parallelogram.

I tried to solve this for awhile. I started writing things that I knew about the situation down, and I found that the (shortest) distance between the plane and either line was constant (though a different constant for both lines). I figured out a few more things but I am not sure if there are necessary for the argument.

Thanks, even the smallest piece of advice would be helpful :-)