The seven circle theorems
Memorise all seven by name. State the name when using.
1. Angle at centre is twice the angle at circumference (same arc).
If a chord subtends an angle of at the centre, it subtends at any point on the major arc.
2. Angle in a semicircle is .
(Special case of theorem 1: if the central angle is , the inscribed angle is .)
3. Angles in the same segment are equal.
Two chords from the same arc subtend the same angle wherever measured on the major arc.
4. Opposite angles in a cyclic quadrilateral sum to .
A 4-vertex inscribed shape has opposite angles supplementary.
5. Tangent is perpendicular to radius at the point of contact.
If a line touches the circle at point , and is the centre, then tangent.
6. Two tangents from an external point are equal in length.
If two tangents are drawn from outside the circle, the segments from external point to tangent points are equal.
7. Alternate segment theorem.
The angle between a tangent and a chord = the angle in the alternate segment (the segment on the other side of the chord).
Worked qualitative. Why is the angle in a semicircle ?
- Diameter creates a central angle of .
- Inscribed angle: .
- This is theorem 2 — special case of 1.
Edexcel tip. ALWAYS state the theorem name. 'Angles in the same segment' or 'angle at centre is twice angle at circumference' — exact phrasing.
- Seven theorems memorised.
- State name when using.
- Apply to find unknown angles.
- Often combined in multi-step problems.