6.5PureStretch

Circles and Triangles

This lesson links triangles to circles via two classic results: the angle in a semicircle is a right angle, and the circumcircle passes through all three vertices of a triangle with its centre at the circumcentre.

25 min Video by Zeeshan Zamurred Circles

Edexcel AS Level Maths: 6.5 Circles and TrianglesWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the “angle in a semicircle” theorem
Recognise when a triangle’s hypotenuse is a diameter
Find the circumcircle of a triangle
Locate the circumcentre using perpendicular bisectors

Angle in a semicircle

If a triangle is drawn inside a circle with one side as a $diameter$ , the angle opposite that diameter is always $90°$ . The converse is just as useful: a right angle means the hypotenuse is a diameter.

diameter as a side \Rightarrow 90° at the opposite vertex

The angle subtended by a diameter is a right angle.

Tip — See a right angle inscribed in a circle? Its hypotenuse is a diameter — so its midpoint is the centre.

The circumcircle

Every triangle has a unique $circumcircle$ passing through all three vertices. Its centre, the $circumcentre$ , is equidistant from the three vertices.

Finding the circumcentre

Because the circumcentre is equidistant from the vertices, it lies on the perpendicular bisector of every side. Find the perpendicular bisectors of two sides; where they meet is the circumcentre. The radius is the distance to any vertex.

1Find the perpendicular bisector of

A B

and of

B C

2Solve them simultaneously to get the circumcentre.

3Radius = distance from the circumcentre to any vertex.

Answercentre = intersection of perpendicular bisectors;

r

= distance to a vertex

Formula recap

∠ = 90° in a semicircle

Diameter subtends a right angle.

circumcentre = perp. bisectors meet

Equidistant from all vertices.

r = distance to any vertex

Circumradius.

Common mistakes to avoid

Assuming any inscribed triangle has a right angle.

Only when one side is a diameter is the opposite angle 90°.

Using only one perpendicular bisector to find the circumcentre.

You need two (then solve simultaneously) to pin down the point.

Key takeaways

A diameter subtends a right angle at the opposite vertex (and vice versa).
The circumcircle passes through all three vertices of a triangle.
The circumcentre is where the sides’ perpendicular bisectors meet; the radius is the distance to a vertex.

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