Basic Properties of Circles

Chords

• The perpendicular from the centre to a chord bisects the chord. (⊥ from centre bisects chord)
• Equal chords are equidistant from the centre. (equal chords, equidistant from centre)

With radius $r$, half-chord $c$ and distance from the centre $d$: $r^2 = c^2 + d^2$.

Angles in a circle

Theorem	Reason to write
Angle at the centre is twice the angle at the circumference on the same arc	∠ at centre twice ∠ at ⊙ce
Angle in a semi-circle is $90°$	∠ in semi-circle
Angles in the same segment are equal	∠s in the same segment
Equal arcs subtend equal angles	equal arcs, equal ∠s

Cyclic quadrilaterals

• Opposite angles add up to $180°$. (opp. ∠s, cyclic quad.)
• An exterior angle equals the interior opposite angle. (ext. ∠, cyclic quad.)

Tangents

• A tangent is perpendicular to the radius at the point of contact. (tangent ⊥ radius)
• Two tangents from an external point are equal in length, and the line from that point to the centre bisects the angle between them. (tangent properties)
• The angle between a tangent and a chord equals the angle in the alternate segment. (∠ in alt. segment)

Common mistake: "∠ in alt. segment" pairs the tangent–chord angle with the angle on the other side of the chord, not the same side.

Exam tip: in a geometric proof, every step needs a reason in brackets. Marks are given for the reasons, not just the final angle.