Can the Midpoint Between Two Equal Circles Ensure Symmetric Area Division?

[ritwik06]

Homework Statement

There are two circles of equal radii. I have to prove that the mid point of the line joining their centres is the only point through which if several arbitrary lines are drawn, equal areas enclosed by the circles will fall on either side of the line.

I cannot think of a way to proceed. I have observed the situation by drawing out equal cirlces and testing the conditions. They seem as obvious as the result of 2+2, but equally difficult to prove.
Please help .

 

[Dick]

Pick a point that is NOT the midpoint. Can you see a way to construct a line through it that does NOT cut the circles generating equal areas? Hint: you may want to consider some special cases. Suppose the point is inside one of the circles? Suppose it's in neither? In the latter case case can you show that if you draw the two lines tangent to one circle that they aren't also tangent to the other? Then think about rotating the line 'a little'.

 

[ritwik06]

Pick a point that is NOT the midpoint. Can you see a way to construct a line through it that does NOT cut the circles generating equal areas? Hint: you may want to consider some special cases. Suppose the point is inside one of the circles? Suppose it's in neither? In the latter case case can you show that if you draw the two lines tangent to one circle that they aren't also tangent to the other? Then think about rotating the line 'a little'.

I just need a formal proof that would be accepted in an exam. I cannot go on with special cases. Thanks for the help though!