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 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.
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 below.
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
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular at a point on a line
- 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)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
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)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary 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
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
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
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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