I've been looking at the paper referenced above, and I'm not convinced - though I stress that I don't have a copy of the SAS macro to verify this - that the solution offered is correct. My reason for saying that is that it looks to me as though it's being assumed that the distances between the centres of the circles are being used as the basis for positioning the circles, as opposed to the areas of the intersections. (For example, if the diagram on the first page is an accurate application of the macro to the numbers given above it, then since the area of circle B is 800 and the area of the intersection AB is 400, circle B should be split into two equal areas by circle A, whereas in fact the circumference of circle A appears to cut circle B directly through its centre - which it shouldn't.)

Years ago when I was a student, a group of us were listening to the radio when the announcer posed a little mathematical puzzle to which he presumably expected a phonecall from a fairly numerate listener with the correct answer within a couple of minutes. The puzzle was this: "A goat is tethered by a rope 10 metres long to the edge of a circular field of grass. He can eat exactly half the grass in the field. What is the radius of the field?" By the time we had been playing with it for the rest of the evening, we came to the conclusion that the problem emphatically wasn't trivial, though we did eventually come up with an iterative solution that converged to the right answer - and it seems to me that this Venn diagram problem is a rather more complicated version of that little conundrum. If so, it might take a little longer to script than you think!

References:

http://mathworld.wolfram.com/GoatProblem.html

http://mathforum.org/library/drmath/view/51746.html