Find the Midpoint of a Segment

This is also called "bisecting a segment," and this construction actually produces the perpendicular bisector of a segment.  It makes use of the theorem which states that the perpendicular bisector of a segment consists of points that are equidistant from the endpoints of the segment.

Given segment AB, construct the perpendicular bisector of that segment.

 

Solution:  Place the point of the compass at A and open it to any convenient length more than half of the segment (but not as long as the segment), and sweep out an arc:

Then place the point of the compass at B, keeping the same radius, and sweep out another arc:

Finally use the straight-edge to draw a line through the two points where the arcs cross:

That line is the perpendicular bisector of segment AB, crossing it at its midpoint, M.

Here is an animation of this process.

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