
Circumcircle of a Triangle
The
circumcircle
of a triangle is the circle that passes through all three vertices of the triangle. The construction first establishes the circumcenter and then draws the circle.
circumcenter
of a triangle is the point where the
perpendicular bisectors
of the sides intersect. This page shows how to construct (draw) the circumcircle of a triangle with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Proof
The image below is the final drawing above with the red labels added.
Note: This proof is almost identical to the proof in
Constructing the circumcenter of a triangle.

Argument 
Reason 
1 
JK is the
perpendicular bisector
of AB. 
By construction. For proof see
Constructing the perpendicular bisector of a line segment 
2 
Circles exist whose center lies on the line JK and of which AB is a
chord. (* see note below) 
The perpendicular bisector of a
chord
always passes through the circle's center. 
3 
LM is the
perpendicular bisector
of BC. 
By construction. For proof see
Constructing the perpendicular bisector of a line segment 
4 
Circles exist whose center lies on the line LM and of which BC is a chord. (* see note below) 
The perpendicular bisector of a
chord
always passes through the circle's center. 
5 
The point O is the circumcenter of the triangle ABC, the center of the only circle that passes through A,B,C. 
O is the only point that lies on both JK and LM, and so satisfies both 2 and 4 above. 
5 
The circle O is the circumcircle of the triangle ABC. 
The circle passes through all three vertices A, B, C 
 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.
Try it yourself
Click here for a printable worksheet containing two triangle circumcircle problems.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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