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Problems on Circles I: Dissecting a Circle

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Given a circle
C
and points A and B, construct a circle passing through A and B and dividing the circumference of
C
into two equal arcs. You can vary the red objects: A, B, the center, and a point on
C
. Note what happens when A is at the center of
C
or when A, B and the center of
C
are aligned.
The circumference of
C
is halved, but is the area of
C
as well?
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