Constructions
146. The Circle That Holds It All · Constructing the circumcircle of a triangle
The circumcentre O is always equidistant from all three vertices.
The circumcircle of a triangle is the unique circle that passes through all three vertices. Its centre — the circumcentre O — is the intersection of the perpendicular bisectors of the three sides, and is equidistant from all three vertices.
What this lesson covers
Try to break it
Drag A, B, or C. The three perpendicular bisectors always meet at one point O, and the circle through O passes through all three vertices. Try to spread the corners so the bisectors miss each other — impossible. Every triangle has exactly one circumcircle.
How you build it
Construct the circumcircle of a triangle.
- Place point A.
- Place point B.
- Place point C.
- Draw side AB.
- Draw side BC.
- Draw side CA.
- Construct the perpendicular bisector of side AB.
- Construct the perpendicular bisector of side AC.
- Mark the intersection of the two bisectors as point O.
- Draw the circumcircle with centre O passing through A.
The proof, step by step
Prove that the circumcentre is equidistant from all three vertices.
- The perpendicular bisector of a segment is the locus of all points equidistant from the segment's endpoints.
- Since O lies on the bisector of AB, OA = OB.
- Since O lies on the bisector of AC, OA = OC.
- Therefore, OA = OB = OC. A circle centred at O with radius OA passes through A, B, and C.
Worked example
In ΔXYZ, the perpendicular bisectors of XY and YZ meet at O. If OX = 8 cm, what is the circumradius of ΔXYZ?
The circumcentre is equidistant from all vertices. Since OX = 8 cm, the circumradius is 8 cm.
- 4 cm
- 8 cm — correct
- 12 cm
- 16 cm