The picture below actually shows the proof.
The goal is to show the red line (which is supposed to bisect circular area) is longer than the diameter AB’.
We see it from:
Length of red curve AB is greater than: AE + EB = AE + EB’
which is certainly longer than AB.
Just one more minute. Let’s review the construction-proof process.
Connect the two endpoints A, B of the curve by a line segment. Draw diameter CD // AB.
Take O (the midpoint of CD), then passing A and O we get another diameter AB’.
The two arguments needed for completing proof are:
(1) Red line must have at least one intersection with diameter CD; (think like this: if the red line resides completely at only one side, then there is no point that it can evenly divide the circular area) Suppose the intersection is at point E;
(2) B and B’ are symmetric to the diameter CD therefore EB = EB’ (think on why? using the property of circles and parallel lines)
Now the rest of the proof is straight forward.
