Circle theorems

#### Tier: #Higher

Description

Circle theorems are a set of rules about angles and lines in circles. You must be able to apply them and give reasons in geometric proofs.

The main theorems:

1. Angle at the centre = twice the angle at the circumference (same arc)
2. Angles in the same segment are equal
3. Angle in a semicircle = 90° (angle subtended by a diameter)
4. Opposite angles in a cyclic quadrilateral sum to 180°
5. Tangent-radius angle = 90° (tangent is perpendicular to the radius)
6. Two tangents from an external point are equal in length
7. Alternate segment theorem: the angle between a tangent and a chord equals the inscribed angle in the alternate segment

Example: If the central angle is $100°$, the inscribed angle on the same arc is: $$\frac{100°}{2} = 50°$$

Always state the theorem used as a reason when answering — do not just write the angle.

Common error: applying the angle-at-centre theorem when the centre is not marked, or confusing "same segment" with "same arc".

Links

Angle facts