Where is the centre?

The idea

The circumcentre lies inside an acute-angled triangle, outside an obtuse-angled triangle, and at the midpoint of the hypotenuse of a right-angled triangle.

Where does O go?

A triangle can be thin and pointy, wide and flat, or have a square corner. Its circumcircle goes through all three corners, and the circumcentre O is the centre of that circle.

Is the circumcentre always inside its triangle?

• Yes, always inside
• No, it can be inside, outside or on a side
• No, it is always on a side

Move C

A and B stay where they are. Drag C anywhere, and watch the circle and its centre O.

Inside, on, outside

The circumcentre lies inside an acute-angled triangle, outside an obtuse-angled triangle, and at the midpoint of the hypotenuse of a right-angled triangle.

Watch the angles. All three below 90° and O is inside. One angle above 90° and O falls outside. One angle exactly 90° and O sits at the midpoint of the longest side.

Notes

The circumcentre is inside an acute-angled triangle, outside an obtuse-angled triangle, and at the midpoint of the hypotenuse of a right-angled triangle.

Check yourself

The angles of a triangle are 50°, 60° and 70°. Where is its circumcentre?

The circumcentre of a triangle lies outside it. What kind of triangle is it?

A triangle has angles 90°, 35° and 55°. Where is its circumcentre?

Triangle ABC is right-angled at B, and its hypotenuse AC is 10 cm. The circumcentre O is the midpoint of AC. What is OA, in cm?

Answer: 5 cm

O is the midpoint of AC, so OA is half of AC: 10 ÷ 2 = 5 cm.

• Outside the triangle. O is outside only when one angle is above 90°. Here all three angles are below 90°.
• At the midpoint of a side. That happens only when one angle is exactly 90°. Here none is.
• Inside the triangle — correct. Yes. All three angles are below 90°, so the triangle is acute-angled and O is inside.
• Obtuse-angled — correct. Yes. One angle is above 90°, and then the circumcentre is outside.
• Acute-angled. In an acute-angled triangle the circumcentre is inside.
• Right-angled. In a right-angled triangle the circumcentre is on the hypotenuse, not outside.
• Inside the triangle. Inside is for acute-angled triangles. This one has a 90° angle.
• At the midpoint of the longest side — correct. Yes. The triangle is right-angled, and its circumcentre is at the midpoint of the hypotenuse.
• Outside the triangle. Outside is for obtuse-angled triangles. This one has a 90° angle.