(
Mathematics  Geometry)
A
circumcenter is the center of
a circle known as the
circumcircle which
circumscribes a
triangle.
It is found at the intersection point of the three perpendicular bisectors of a triangle, and also lies on the
Euler line.
Claim
Given triangle
ABC, the
perpendicular bisectors are
concurrent at the
circumcenter.
Proof
Let line
l be the
perpendicular bisector of
AB,
let line
m be the
perpendicular bisector of
BC, and
let
D be the
intersection point of
l and
m.
Since a point is on a
perpendicular bisector of two points
if and only if that point is equidistant from the pair of points, it follows that since
D is on
l,

D  A = 
D 
B and since
D is on
m, 
D 
B = 
D 
C.
Consequently, 
D 
A = 
D 
C.
D is equidistant from
A and
C, therefore
D is on the
perpendicular bisector of
AC. All three lines share
D and so the lines are
concurrent at
D.
D is
equidistant from all three corners. Hence
D is the
circumcenter.
Uniqueness
Since the circumcenter is
equidistant from
A,
B and
C, it must lie on both lines
l and
m. Lines
l and
m cannot be paralell by the definition of
triangle, so the two lines
intersect at one point only, proving the uniqueness of the circumcenter.
See also other triangle centers:
incenter,
centroid,
orthocenter.