Constructing the Midpoint of a Line Segment

Constructing the Midpoint of a Line Segment

How to construct the midpoint of a line segment (and the perpendicular bisector of a line segment).

Goal

Constructing the Midpoint of a Line Segment: Goal

Goal:
Constructing the red point
the midpoint of the given line segment

How to Construct

Constructing the Midpoint of a Line Segment: Step 1

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

Draw two arcs at the upper part
and the lower part of the segment.
(red arcs)

The radius of the compass has to be larger than
one half the length of the segment.

Constructing the Midpoint of a Line Segment: Step 2

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

Draw two arcs (red arcs)
that intersect the previous arcs (blue arcs).

Constructing the Midpoint of a Line Segment: Step 3

Draw a line that passes through
the intersecting points formed by the arcs.

Then the intersection of the vertical line and the line segment is the midpoint of the line segment. And the vertical line is the perpendicular bisector of the line segment.

Then the intersection
of the vertical line and the line segment
is the midpoint of the line segment.
(red point)

And the vertical line
is the perpendicular bisector of the line segment.
(blue line)