The circle around a triangle

The idea

The circle through the vertices of a triangle is its circumcircle, which circumscribes the triangle (the triangle is inscribed in it); its centre, the circumcentre, is where the perpendicular bisectors of the sides meet.

A lid for a triangle

Cut a triangle out of paper. Can you find a round lid that touches all three corners at once?

Draw the perpendicular bisector of each of the three sides of a triangle. What happens?

• All three lines pass through one point
• They make a small triangle in the middle
• Only two of them ever meet

Three bisectors

Tap a side to draw its perpendicular bisector. Then draw the circle.

One point, one circle

The circle through the three vertices of a triangle is its circumcircle, and its centre is the circumcentre: the point where the perpendicular bisectors of the sides meet. The circle circumscribes the triangle, and the triangle is inscribed in the circle.

O is on the bisector of AB, so OA = OB. It is on the bisector of BC, so OB = OC. So OA = OB = OC, and a circle centred at O passes through all three corners. The third bisector goes through the same point O.

Notes

The circle through the vertices of a triangle is its circumcircle; its centre, the circumcentre, is where the perpendicular bisectors of the sides meet.

Check yourself

The three vertices of triangle ABC lie on a circle. What do we say about the triangle?

O is the circumcentre of triangle ABC. Which statement is true?

O is the circumcentre of triangle ABC and OA = 3 cm. What is the diameter of the circumcircle, in cm?

Answer: 6 cm

OA is a radius, and the diameter is twice the radius: 2 × 3 = 6 cm.

The perpendicular bisectors of AB and BC meet at O. Which other line must also pass through O?

• ABC circumscribes the circle. It is the other way round. The circle circumscribes the triangle, and the triangle is inscribed in the circle.
• ABC is inscribed in the circle — correct. Yes. The triangle is inscribed in the circle, and the circle circumscribes the triangle.
• ABC is the circumcircle. The circumcircle is the circle itself, not the triangle.
• O is the midpoint of AB. O is equally far from A and B, but it is not the midpoint of AB unless the triangle is special.
• OA + OB = OC. The three distances are equal: OA = OB = OC.
• OA = OB = OC — correct. Yes. O is on all three perpendicular bisectors, so it is equally far from A, B and C.
• The perpendicular bisector of CA — correct. Yes. All three perpendicular bisectors of a triangle meet at the circumcentre.
• The side CA itself. The side CA is a different line. The line through O is the perpendicular bisector of CA.
• The side AB itself. The side AB is a different line. The line through O is the perpendicular bisector of AB.