I think I can get eleven. Start with drawing a diameter, and then another diameter that is the first's perpendicular bisector. Continue by drawing a line 45 degrees tilted from the first diameter but shift it slightly down to prevent it from crossing the center. Now draw a line tilted about 70 degrees from the first diameter and again lower it so that it crosses all three past lines making eleven sections. (These don't have to be exact).
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Yeah, this also seems a possible solution - actually this was the one I had in mind...