Go directly to content
Search site >



Circumcenter of a Triangle
Geometry construction using a compass and straightedge
Step-by-step Instructions Printer friendly version

This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.

After doing this Your work should look like this

We start with a triangle ABC.

1.  Find the bisector of one of the triangle sides. Any one will do.

See Constructing the Perpendicular Bisector of a Line Segment for detailed instructions.

2.  Repeat for the another side. Any one will do.
3.  Mark the point where these two perpendiculars intersect as point O.
(Optional step) Repeat for the third side. This will convince you that the three bisectors do, in fact, intersect at a single point. But two are enough to find that point.

Done   The point O is the circumcenter of the triangle ABC.

Note: This point may be outside the triangle. This is normal.



Note: The circumcenter is the center of a triangle's circumcircle, and the construction of the circumcircle is almost the same as this one, with the addition of the last step to actually draw the circle.

See Constructing the circumcircle of a triangle.

Proof

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

  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 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.

  - 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 circumcenter 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.

Constructions pages on this site

Lines

Angles

Triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

(C) 2009 Copyright John Page