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.
Wolfram
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