A geometric construction done with a movable compass alone. All constructions possible with a compass and
straightedge are possible with a movablecompass alone, as was proved by Mascheroni (1797). Mascheroni's
results are now known to have been anticipated largely by Mohr (1672).
An example of a Mascheroni construction of the midpoint of a line segment specified
by two points
and
illustrated above (Steinhaus 1999, Wells 1991). Without loss of generality, take
.
1. Construct circles centered at and passing through and . These are unit circles centered at (0, 0) and (1, 0).
2. Locate ,
the indicated intersection of circles and , and draw a circle centered on passing through points and . This circle has center (1/2, ) and radius 1.
3. Locate ,
the indicated intersection of circles and , and draw a circle centered on passing through points and . This circle has center (3/2, ) and radius 1.
4. Locate ,
the indicated intersection of circles and , and draw a circle centered on passing through point . This circle has center (2, 0) and radius .
5. Locate
and ,
the intersections of circles and . These points are located at positions (5/4, ).
6. Locate ,
the intersection of circles and . This point has position (1/2, 0), and is therefore the desired
midpoint of .
Pedoe (1995, pp. xviii-xix) also gives a Mascheroni solution.
of a line segment specified
by two points 
and 
illustrated above (Steinhaus 1999, Wells 1991). Without loss of generality, take
.
and 
passing through 
and 
. These are unit circles centered at (0, 0) and (1, 0).
,
the indicated intersection of circles 
and 
, and draw a circle centered on 
passing through points 
and 
. This circle has center (1/2, 
) and radius 1.
,
the indicated intersection of circles 
and 
, and draw a circle centered on 
passing through points 
and 
. This circle has center (3/2, 
) and radius 1.
,
the indicated intersection of circles 
and 
, and draw a circle centered on 
passing through point 
. This circle has center (2, 0) and radius 
.
and 
,
the intersections of circles 
and 
. These points are located at positions (5/4, 
).
,
the intersection of circles 
and 
. This point has position (1/2, 0), and is therefore the desired
midpoint of 
.
