Problems on Circles I: Dissecting a Circle

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

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