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.

We now show the possible set of first lines that can be drawn from the intersection points of the two circles.

No two of these lines will construct the point F. So we must construct the second line using new points arising from the intersection of a circle and a line. If the second line is to construct the desired point F on AB, it cannot go through a point on AB. But every new line goes through points already on the line AB or is parallel to AB.

This indicates why we cannot trisect the segment using only two circles and two lines.

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Villanova Home Page Math Dept Home Page 8 Feb 2001