Construction: Midpoint of a Line Segment

How to construct the midpoint of a line segment (and the perpendicular bisector): by using a compass and a straightedge (ruler).

Example

Given

Let's see how to construct
the midpoint of the given line segment.

Construction

Place the point of the compass
at the left endpoint of the line segment.
(blue point)

Draw two arcs
on the top and bottom of the segment.

Set the radius of the compass
that seems to be larger than
one half the length of the segment.

Without changing the radius of the compass,
place the point of the compass
at the other endpoint of the line segment.

Draw two arcs
on the top and bottom of the segment.

Draw a line that passes through
the intersecting points
of the top arcs and the bottom arcs.

Then the intersection
of the vertical line and the given line segment
is the midpoint of the line segment.

Perpendicular Bisector

And the blue line
is the perpendicular bisector of the line segment.