We first consider the two ways we can construct two circles. The second case does not allow any new lines to be drawn so may be eliminated. For the case with two congruent circles, by symmetry we only need consider lines through the "first quadrant:": through the point C, D or E. The only possibilities are the blue lines. The only new points created are G, H, I, J. The only new lines that go through one of these new points and the original points off the x-axis are the two red lines which are parallel to the x-axis. In any case, the only constructible points on the x-axis are integral or half-integral.
Villanova Home Page Trisection Index of Pages 2 July 2001