Circle theorems

Edexcel 1MA1 Higher papers always include at least one circle theorem question — typically a multi-step problem worth 4–6 marks requiring you to state reasons.

The seven standard Edexcel circle theorems

Theorem 1 — Angle at the centre: The angle subtended at the centre is twice the angle subtended at the circumference by the same arc. ∠AOB = 2 × ∠ACB (where O is the centre, A, B, C are on the circle).

Theorem 2 — Angle in a semicircle: The angle in a semicircle is 90°. (Special case of Theorem 1 — the arc is a semicircle.) If AB is a diameter, then ∠ACB = 90° for any point C on the circle.

Theorem 3 — Angles in the same segment: Angles subtended by the same chord at the same side of the circle are equal. ∠ADB = ∠ACB (both subtended by chord AB from the same segment).

Theorem 4 — Cyclic quadrilateral: Opposite angles in a cyclic quadrilateral (all four vertices on the circle) sum to 180°. ∠A + ∠C = 180°; ∠B + ∠D = 180°.

Theorem 5 — Tangent–radius: A tangent to a circle is perpendicular to the radius at the point of contact. ∠OAT = 90° where OA is a radius and AT is a tangent.

Theorem 6 — Two tangents from an external point: Two tangents drawn from an external point are equal in length. PA = PB where P is external and A, B are the points of contact.

Theorem 7 — Alternate segment theorem: The angle between a tangent and a chord equals the angle in the alternate segment. ∠TAB = ∠ACB (angle in alternate segment).

Proof requirement (Edexcel mark scheme)

For circle theorem angle-chasing, you MUST state the theorem name (or a clear description) as a reason for each step. Writing numbers alone, without reasons, loses marks.

Acceptable reasons: "angle at centre is twice angle at circumference", "angles in same segment are equal", "opposite angles in cyclic quad sum to 180°", "tangent perpendicular to radius", "alternate segment theorem".