Best Answer:
Well, here's what I got:

A............B......C

___________

\................\........\

.\................\........\

..\_______\____\

....D............E......F

So the parallelogram ACFD is the original, and BCFE is the smaller but similar one with which it shares 2 corners. now if you're trying to use a parallelogram that shares ONLY one corner, that's pretty complicated - unless the angle DFC is pretty acute, B won't touch line AC but would instead be somewhere in the middle of ACFD.

anyway, taking it to be the case that B also lies on AC, here's a few simple things we know:

The proportion of AC or DF to AD, BE, or CF is equal to the proportion of AD, BE, or CF to BC or EF.

AB or DE plus BC or EF is equal to AC or DF

you could also figure out all sorts of fun stuff if you knew the height of either parallellogram or any of the angles.

Source(s):
by the way, ascii drawings are VIRTUALLY IMPOSSIBLE in this stupid format! the way yahoo answers switches from font to font between your original post, your preview, and the final product... INFURIATING! >B-O