http://www.forrestrace.com/half-circle/
Prove that a curve with endpoints on a unit circle and which divides the circle in half has length ≥ 2?
Prove that a curve with endpoints on a unit circle and which divides the circle in half (two pieces of equal area) has length of at least 2.
PINKGREEN
I cannot agree that any curve which divides the circle in half must pass through the origin.
For such a curve in the circle of length less than 2 (so that it starts and ends on the perimeter) there exists a diameter that doesn’t intersect the curve.
Proof: In order for the curve to intersect all the diameters inside the circle, it must either contain the origin or wrap around it. But the minimal distance from the origin and back to starting point is 2. Therefore, a curve with length less than 2 cannot contain the origin, nor can it wrap around it.
Then it’s clear that after drawing this diameter, we have the curve “only on one side of the circle” meaning it’s dividing the half circle. Hence it’s impossible for it to cut the circle into two equal areas.
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