Dividing with Straightedge
Given a line segment and a line parallel to it, divide the segment in three equal parts using only a straightedge (a ruler with no markings). You can draw straight lines, but not measure distances.
Draw two intersecting lines through the ends of the segment. Draw the diagonals of the trapezoid formed by those lines, the segment, and the parallel line. The line through the intersection of the diagonals will divide both bases of the trapezoid in two equal parts.
Repeat the same operation again with each half to split the base in four equal parts. Use three of the four parts to split the original segment in three.