This line is the perpendicular bisector. The point of intersection of the perpendicular bisector and the original line is exactly halfway along the first line. If everything has been constructed ...

In the diagram below, AB is the chord of a circle with centre O. OM is perpendicular to AB (meet at a right angle). Look at triangles OAM and OBM. The hypotenuses (OA and OB) are the same ...