There may be several such lines. Here's an outline of the construction. A n is the desired polygon. Pick up a point A. Rotate the segment AA 1 in M 1 through the angle a 1 . Since the rotation is a motion of the plane that preserves shapes and distances, the image of the segment AA 1 is equal in length to the segment itself.
All intermediate segments have equal lengths.
The last one again ends at A 1 . Let its other end be A'. This means that A 1 lies on the locus of points equidistant from A and A' - the perpendicular bisector of the segment AA'. Pick up another point B and similarly construct B'. The perpendicular bisectors of AA' and BB' are bound to intersect at A 1 .