Assume the initial circle is centered at (0,0) and has radius 1. Now draw a pair of circles of half that radius, one centered at (0, 1/2), the other at (1/2, 0). These intersect (in the sense of touching at a single point) at (0,0), and thus subdivide the large circle into 4 equalarea subsets, two of which are circles, two of which are not.
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Originally Posted by Wacky
Next divide the remainder into 3 equal parts.
The easiest way would be to use radial lines. But this is not allowed.
Therefore simply rotate the outer endpoints a uniform distance along the circumference.

"Outer endpoints" of what?
It seems to me a tricky problem might be to subdivide into 4 equalarea pieces, without having any of the resulting piece boundaries intersect the center of the original circle, and without ever placing the point of one's compass or ruler so it touches the center. I need to think about that a bit more...