I'll make it plane
Geometry
Level
1
Let two shapes both topologically equivalent to a circle be drawn on the same plane.
True or false?
There exists a line on the plane that exactly bisects both shapes (in terms of area).
True
Insufficient information provided to determine
False
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1 solution
Take the first shape and consider all the lines that bisect it. There must be one of these lines at any angle you choose. (Proof: draw said line at the angle in question and move it across the plane until it is tangent to the shape, then continue to move it until it has passed through and is again tangent to the shape. If you consider the percentage of the shape on the upper side of the line, it started at 100% and finished at 0% or vice versa, so at some point during the traverse it must have been at exactly 50%.)
Now consider the second shape. As we change the angles of the lines that bisect the first shape, one of them must be tangent to the second shape. Continue to rotate the bisector until it is tangent on the other side of second shape. By the same argument we used for the first shape, since the percentage of the second shape on one side went from 0 to 100%, it must at some point have been 50%. We now have a line that cuts off 50% of both shapes.
(In three dimensions, with a planar cut that bisects three arbitrary figures in 3-D space, this is known as the ham sandwich theorem, because you can always find a cut that divides both the two pieces of bread and the slice of ham exactly in half.)