Originally posted by doodinthemood
Any line which goes through the centre of a pancake cuts it in half. Thus, any line that cuts through the centres of both pancakes cuts both of them. In any situation, it is possible to get a straight line between two points so it is always possible.
let me rephrase... these pancakes are not by definition circular. in fact they could be very oddly shaped blobs. choosing them to be convex was a way of trying to avoid arguments about discontinuous blobs, but that may have been misleading insofar as making this seem simpler and more obvious than it is.
imagine you've got a couple of weird shaped blobs on a plate, but no holes in the middle of the blobs. prove that you can cut them both in half with a single cut of a big ol' knife (you can even bring out your katana, if you so choose).
note: with oddly shaped figures, you could get vastly different results in terms of area splitting, by cutting though a particular point in the figure at a different angle. i.e. a straight up and down cut through the centroid could be way different from a horizontal cut or a slight angle or a 45, etc.