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 
.
