A perpendicular bisector 
of a line segment 
is a line segment perpendicular
to 
and passing through the midpoint 
of 
(left figure). The perpendicular bisector of a line segment
can be constructed using a compass by drawing circles
centered at 
and 
with radius 
and connecting their two intersections. This line segment crosses 
at the midpoint 
of 
(middle figure). If the midpoint 
is known, then the perpendicular bisector can be constructed
by drawing a small auxiliary circle around 
, then drawing an arc from each endpoint that crosses the line
at the farthest intersection of the circle with the line (i.e., arcs with radii 
and 
respectively). Connecting the intersections of the arcs then gives the perpendicular
bisector 
(right figure). Note that if the classical construction requirement that compasses
be collapsible is dropped, then the auxiliary circle can be omitted and the rigid
compass can be used to immediately draw the two arcs using any radius larger that
half the length of 
.
The perpendicular bisectors of a triangle 
are lines passing through the midpoint 
of each side which are perpendicular to the given
side. A triangle's three perpendicular bisectors meet
(Casey 1888, p. 9) at a point 
known as the circumcenter
(Durell 1928), which is also the center of the triangle's circumcircle.
