Circle theorems

🔗What you need to know first

How to

Circle theorems are a set of rules about angles and lines in circles. You must be able to apply them and give reasons in geometric proofs.

The main theorems:

Angle at the centre = twice the angle at the circumference (same arc)
Angles in the same segment are equal
Angle in a semicircle = 90° (angle subtended by a diameter)
Opposite angles in a cyclic quadrilateral sum to 180°
Tangent-radius angle = 90° (tangent is perpendicular to the radius)
Two tangents from an external point are equal in length
Alternate segment theorem: the angle between a tangent and a chord equals the inscribed angle in the alternate segment

Example: If the central angle is $100°$, the inscribed angle on the same arc is: $$\frac{100°}{2} = 50°$$

Always state the theorem used as a reason when answering — do not just write the angle.

Common error: applying the angle-at-centre theorem when the centre is not marked, or confusing "same segment" with "same arc".

Common topics that go alongside

❓Questions to practise

New to Bow Tie Maths? It generates questions on this topic, marks them instantly, and tracks what you've mastered. Free to sign up.

📝Past paper questions

💬What the examiners say

"You need to state what the theorem says, for example 'angles in the same segment are equal' rather than just 'same segment theorem'."
"As it was a proof, it required reasons to be given at each stage, and although a good number knew the facts, they did not give correct, or complete reasons and struggled to gain any credit. It is important again that students practise stating the geometric properties and reasons."
"Centres may wish to reinforce the importance of using accurate mathematical language in justifications."

⬆️How you can quickly improve

Learn the full wording of each circle theorem and practise writing them from memory — the precise statement, not just the name.
Before choosing a theorem, check the diagram carefully: are the angles at the circumference or the centre? Is there a tangent? Are all four vertices on the circle?
Write each step of a proof on a separate line with both the angle value and the complete geometric reason alongside it.

💡Watch

🔓What this unlocks

ℹ️Calculator tricks