Circle Theorems Cambridge (CIE) IGCSE Maths Extended

Angles at Centre and Circumference

  • The angle at the centre is twice the angle at the circumference when subtended by the ends of the same arc or chord.
  • This theorem applies to shapes that look like an arrowhead or a diamond.
  • When the lines form a diamond, compare the reflex angle at the centre with the angle at the circumference.

Angle in a Semicircle

  • The angle in a semicircle is 90∘90^{\circ}.
  • Lines drawn from a point on the circumference to either end of a diameter are perpendicular.
  • This is a specific instance of the centre/circumference theorem where the angle on the diameter is 180∘180^{\circ}.

Theorems with Chords

  • A chord is a straight line joining any two points on the circumference.
  • The perpendicular bisector of a chord passes through the centre of the circle.
  • A radius bisects a chord at right angles (90∘90^{\circ}).
  • Chords of equal length are equidistant from the centre.

Theorems with Tangents

  • A tangent touches the circumference at exactly one point and is external to the circle.
  • A radius and a tangent meet at right angles (90∘90^{\circ}).
  • Tangents from the same external point are equal in length and can form a kite with the radii.
  • This kite structure comprises two congruent triangles and has two right angles where tangents meet the radii.

Angles in Cyclic Quadrilaterals

  • A cyclic quadrilateral is formed by four points that all lie on the circumference.
  • Opposite angles in a cyclic quadrilateral add up to 180∘180^{\circ}.

Angles in the Same Segment

  • Angles in the same segment are equal.
  • Any two angles on the circumference formed from the same two points (the same chord) are equal.
  • This often appears as a "bowtie" shape in geometric diagrams.

The Alternate Segment Theorem

  • The angle between a chord and a tangent is equal to the angle in the alternate segment.
  • To identify equal angles, look for a cyclic triangle where one vertex meets a tangent; the angle between the tangent and the triangle side equals the interior angle at the opposite vertex.