The eight circle theorems

OCR J560 Higher (J560/04–06) tests the standard set of circle theorems. Each can either be applied to find an angle, or proved from first principles. Memorise both.

The theorems

1. Angle at the centre = 2 × angle at the circumference (subtended by the same arc). If arc AB subtends angle 2θ at centre O and angle θ at circumference point P, then ∠AOB = 2∠APB.

2. Angle in a semicircle = 90°. Special case of (1) with the central angle = 180°.

3. Angles in the same segment are equal. Two angles at the circumference subtended by the same arc are equal.

4. Opposite angles of a cyclic quadrilateral sum to 180°. A cyclic quadrilateral has all four vertices on the circumference. Opposite angles are supplementary.

5. Tangent ⊥ radius at the point of contact.

6. Two tangents from a point are equal in length.

7. Alternate segment theorem. The angle between a tangent and a chord at the point of contact equals the angle in the alternate segment.

8. Perpendicular from the centre to a chord bisects the chord.

How to use them in a problem

Look at every angle/length stated, then ask: which theorem applies? Often you need two or three in sequence.

Worked example. A, B, C, D lie on a circle. ∠ABC = 110°. What is ∠ADC? ABCD is a cyclic quadrilateral. By theorem 4, ∠ABC + ∠ADC = 180°. So ∠ADC = 180° − 110° = 70°.

Proofs (Higher)

OCR rewards rigorous reasoning. Every step should cite a theorem or basic fact.

Prove angle in a semicircle = 90°. Let A, B be ends of a diameter (centre O). Let P lie on the circle, P ≠ A, B. In triangle OAP, OA = OP (radii), so ∠OAP = ∠OPA. Call this α. Similarly in triangle OBP: ∠OBP = ∠OPB. Call this β. The full angle ∠APB = α + β. The angles in triangle ABP sum to 180°: α + β + ∠APB = 180° → 2(α + β) = 180° → α + β = 90°. So ∠APB = 90°. ∎

OCR mark scheme conventions

• Each step that uses a named theorem must say so: "(angle at centre = 2 × angle at circumference)" or just the abbreviated reason.
• "Reasons" are mandatory in proof questions — answers without reasons score at most 1/4.
• A diagram with key angles labelled is acceptable working.