Geometry: Circle Theorems

Circle theorems describe key angle and length relationships in circles. At GCSE you are expected to know, apply and cite all eight theorems, and in some cases prove them.

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Circle Vocabulary (Review)

Term	Definition
Centre	The fixed middle point
Radius (r)	Distance from centre to any point on the circle
Diameter (d)	Chord through the centre; d = 2r
Chord	A straight line joining two points on the circle
Arc	Part of the circumference
Sector	Region bounded by two radii and an arc
Tangent	A line touching the circle at exactly one point
Segment	Region between a chord and an arc
Cyclic quadrilateral	A quadrilateral with all four vertices on the circle

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The Eight Circle Theorems

Theorem 1: Angle at the Centre

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

Angle AOB = 2 × angle ACB (where O is the centre)

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Theorem 2: Angle in a Semicircle

The angle in a semicircle is always 90°.

If AB is a diameter, then angle ACB = 90° for any point C on the major arc.

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Theorem 3: Angles in the Same Segment

Angles subtended by the same chord, from the same side of the chord (in the same segment), are equal.

Angle ACB = angle ADB (both stand on arc AB, same segment)

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Theorem 4: Cyclic Quadrilateral

Opposite angles in a cyclic quadrilateral sum to 180° (they are supplementary).

Angle A + angle C = 180°; Angle B + angle D = 180°

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Theorem 5: Tangent–Radius

A tangent to a circle is perpendicular to the radius at the point of contact.

Angle OPT = 90° (O = centre, P = point of tangency, T = point on tangent)

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Theorem 6: Two Tangents from an External Point

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

If PA and PB are tangents from external point P: PA = PB

Consequence: triangle OAP ≡ triangle OBP (RHS congruence).

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Theorem 7: Perpendicular from Centre to Chord

The perpendicular from the centre of a circle to a chord bisects the chord (divides it into two equal halves).

If OM is perpendicular to chord AB: AM = MB

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Theorem 8: Alternate Segment Theorem