Circle theorems

Circle theorems are properties of angles and lines within and around circles. CCEA Higher tier questions on circle theorems are common and require both calculating angles and writing geometric reasons.

The eight circle theorems

Theorem 1: Angle at the centre

The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.

If angle at centre = 2θ, then angle at circumference = θ.

Theorem 2: Angles in the same segment

Angles subtended by the same arc at the circumference are equal.

All inscribed angles that subtend the same chord (from the same side) are equal.

Theorem 3: Angle in a semicircle

The angle in a semicircle (angle subtended by a diameter at the circumference) is 90°.

If AB is a diameter and C is on the circle, then angle ACB = 90°.

Theorem 4: Cyclic quadrilateral

Opposite angles in a cyclic quadrilateral (all vertices on a circle) add up to 180° (they are supplementary).

Theorem 5: Tangent perpendicular to radius

A tangent to a circle is perpendicular to the radius at the point of contact. Angle = 90°.

Theorem 6: Two tangents from an external point

The two tangents drawn from an external point to a circle are equal in length.

Theorem 7: Alternate segment theorem (tangent-chord angle)

The angle between a tangent and a chord at the point of tangency equals the inscribed angle on the opposite side of the chord.

Theorem 8: Perpendicular from centre bisects chord

The perpendicular from the centre of a circle to a chord bisects the chord.

Proof structure for CCEA

CCEA Higher questions often say "Give a reason for each step." You must name the theorem, not just state the answer. Use exact names:

• "Angle at the centre is twice the angle at the circumference"
• "Angles in the same segment are equal"
• "Opposite angles in a cyclic quadrilateral sum to 180°"
• "Tangent-radius is perpendicular"
• "Alternate segment theorem"