Key Points

Equal chords of a circle subtend equal angles at the centre. Conversely, if the angles subtended by two chords at the centre are equal, then the chords are equal. If chord $AB = CD$, then $\angle AOB = \angle COD$ and vice-versa.

A perpendicular line drawn from the centre of a circle to a chord bisects the chord. If $OM \perp AB$ where $O$ is the centre and $AB$ is the chord, then $AM = MB$.

The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. This is the converse of the previous theorem. If $M$ is the midpoint of chord $AB$, then $OM \perp AB$.

Equal chords of a circle are equidistant from the centre. Conversely, chords that are equidistant from the centre of a circle are equal in length.

In a circle, if two chords are equal, then their corresponding arcs (minor and major) are congruent. Conversely, if two arcs are congruent, their corresponding chords are equal.

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. For arc $PQ$, $\angle POQ = 2 \times \angle PAQ$.

Angles subtended by the same arc (or chord) at any points on the circumference in the same segment are equal. If points $C$ and $D$ are in the same segment, $\angle ACB = \angle ADB$.

The angle in a semicircle is always a right angle ($90^\circ$). If $AB$ is a diameter and $C$ is any point on the circle, then $\angle ACB = 90^\circ$.

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line, then the four points lie on a circle (they are concyclic).

The sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$. For a cyclic quadrilateral $ABCD$, $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.

If the sum of a pair of opposite angles of a quadrilateral is $180^\circ$, then the quadrilateral is cyclic. This is the converse of the cyclic quadrilateral property.

Congruent arcs of a circle subtend equal angles at the centre. If arc $AB$ is congruent to arc $CD$, then $\angle AOB = \angle COD$.