Problems on Circles I: Dissecting a Circle

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?