Prove there is a line that cuts the area in half Suppose you are given two compact, convex sets A and B in the plane, prove there exists a line such that the area of A and B is simultaneously divided in half.
Can you help me with this proof? What I think I have to do is give a fuction that measures the area and then use  intermediate value theorem, but I don't know how to give this function explicitly, I'm a third semester undergrad, so I cannot use very advanced tools.
Thanks for any suggestions.

Okay, this is convoluted so bear with me.
For any point x in the plane you can construct a line, L, from the origin of the plane through x.  For each real number r (positive or negative) you can construct a line perpendicular to L that is r distance from the origin.  Precisely one of these lines at a specific distance r from the origin will cut A is half.  (Basically, by the intermediate value theorem by considering a function that measures what portion of A is cut by the perpendicular line r distance from the origin.  The is contionous and goes from 0 to 100% and is monotonic so there is exactly one, r, where the result is 50%.)
Thus we can define f: Plane -> Real Numbers where f(x) is the real number r where the perpendicular line cuts A in half.  This is a continuous function.
Define g: Plane -> Real Numbers where g(x) is the real number r where the perpendicular line cuts B in half.
Define h(x) = f(x) - g(x).
By the fixed point theorem there must be a point y in the plane where h(y) is 0.  Then f(y) = g(x) = some real number r.  Find the line that is that is r distance from the origin and perpendicular to the line from y to the origin.
This line will cut both A and B in half.
