After doing this	Your work should look like this
We start with three given points A,B,C. Goal: Construct a circle that passes through all three.
1.  (Optional*) Draw straight lines to create the line segments AB and BC. Any two pairs of the points will work. * We draw the two lines to make it clear when we later draw their perpendicular bisectors, but it is not strictly necessary for them to actually be there to do this.
2.  Find the perpendicular bisector of one of the lines. See Constructing the Perpendicular Bisector of a Line Segment.
3.  Repeat for the other line.
4.  The point where these two perpendiculars intersect is the center of the circle we desire.
5.  Place the compass point on the intersection of the perpendiculars and set the compass width to one of the points A,B or C. Draw a circle that will pass through all three.
6.  Done. The circle drawn is the only circle that will pass through all three points.

Explanation

This is virtually the same as constructing the circumcircle a triangle. If you draw three lines linking the given points, you will get a triangle. The circumcircle passes through all three vertices, just as here.

Proof

The image below is the final drawing above with the red items added.

  - Q.E.D

* Note
Depending where the center point lies on the bisector, there is an infinite number of circles that can satisfy this. Two of them are shown on the right. Steps 2 and 4 work together to reduce the possible number to just one.

Constructions pages on this site

Lines

• Introduction to constructions
• List of printable constructions worksheets
• Copy a line segment
• Perpendicular bisector of a line segment
• Perpendicular from a line at a point
• Perpendicular from a line through a point
• Perpendicular from endpoint of a ray
• Divide a segment into n equal parts
• Parallel line through a point (angle copy method)
• Parallel line through a point (rhombus method)

Angles

• Bisecting an angle
• Copy an angle
• Construct a 30° angle
• Construct a 45° angle
• Construct a 60° angle
• Construct a 90° angle (right angle)
• Constructing  75°  105°  120°  135°  150° angles and more

Triangles

• Copy a triangle
• Isosceles triangle, given base and side
• Isosceles triangle, given base and altitude
• Equilateral triangle
• Triangle medians
• 30-60-90 triangle, given the hypotenuse
• Triangle, given 3 sides (sss)
• Triangle, given one side and adjacent angles (asa)
• Triangle, given two sides and included angle (sas)
• Incircle of a triangle
• Circumcircle of a triangle

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the center of a circle
• Circle given 3 points
• Tangent at a point on the circle
• Tangents through an external point
• Focus points of a given ellipse

Polygons

• Hexagongiven one side
• Hexagon inscribed in a given circle
• Pentagon inscribed in a given circle

Non-Euclidean constructions

• Construct an ellipse with string and pins
• Find the center of a circle with any right-angled object