Given a Segment, Construct Its Perpendicular Bisector; Given a Triangle, Construct Its Circumcircle
This Demonstration shows two constructions:[more]
1. The perpendicular bisector of a segment .
2. The circumcircle of a triangle .
The second construction uses the first construction twice.
Construct the perpendicular bisector of and of . The center of the circumcircle of the triangle is the intersection of these bisectors.
1. Draw the line segment .
2. Draw two circles with the same radius and centers and . (Any radius works as long as the circles intersect at two points.)
3. The circles intersect at two points, and . The perpendicular bisector of is the line through these two points. The point is the midpoint of .
1. Draw a triangle .
2. Draw two perpendicular bisectors of —for example, of and . Let be the intersection of the two bisectors.
3. The circumcircle has center and radius .[less]
Euclid I. 10. Construct the midpoint of a given segment.
Euclid IV. 5. About a given triangle, circumscribe a circle.
 G. E. Martin, Geometric Constructions, New York: Springer, 1998, pp. 4–5.