It is impossible because a "line", technically goes on for infinity in both directions. A line goes on forever, no midpoint. A "line segment" however, can be bisected.
Bisection and trisection involve cutting something into two or three equal parts. The dividing line is called the "bisector". After you do the bisecting or trisecting, you end up with congruent parts of the segment or angle you cut up.
Segment bisection, the related term midpoint, and segment trisection are pretty simple ideas. (Their definitions, which follow, are used frequently in proofs.)
Segment bisection: A point, segment, ray, or line that divides a segment into two congruent segments bisects the segment.
Midpoint: The point where a segment is bisected is called the midpoint of the segment; the midpoint cuts the segment into two congruent parts.
Segment trisection: Two things (points, segments, rays, lines, or any combination of these) that divide a segment into three congruent segments trisect the segment. The points of trisection are called — check this out — the trisection points of the segment.
Learn more about bisection and trisection on Dummies.com.