How do you contrust the mid point of a given line?

First, it needs to be a line segment: a line is infinitely long and so has no midpoint.

Suppose the line to be bisected is AB. Place the point of a pair of compasses at A with an arc which is greater than half AB. Draw arcs above and below the line segment. Then, move the compass point to B and without changing the arc width, draw fresh arcs to intercept the previous ones at points X and Y. The intersection of the line segment XY and AB is the midpoint of AB.