Circumcenter of a Triangle
Geometry construction using a compass and straightedge

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is also the center of the circumcircle, the circle that passes through all three vertices of the triangle. This page shows how to construct (draw) the circumcenter 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.

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.

## Printable step-by-step instructions

The above animation is available as a printable step-by-step 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.

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.

* 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.
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