Key Points

A circle is the set of all points in a plane that are at a fixed distance (radius, $r$) from a fixed point (centre).

A chord is a line segment that joins any two points on a circle. The diameter is the longest chord, passes through the centre, and its length is twice the radius ($d=2r$).

An infinite number of circles can pass through two given points. Exactly one unique circle can pass through three non-collinear points.

The line drawn perpendicular from the centre of a circle to a chord bisects the chord. Conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord.

The length of a chord is given by the formula $2\sqrt{r^2 - d^2}$, where $r$ is the radius and $d$ is the perpendicular distance from the centre to the chord. This is an application of the Pythagorean theorem.

In a circle, chords of equal length subtend equal angles at the centre. The converse is also true: if chords subtend equal angles at the centre, they are equal in length.

Chords of equal length in a circle are equidistant from the centre. The converse is also true: chords that are equidistant from the centre are equal in length.

Of two unequal chords in a circle, the longer chord is closer to the centre than the shorter chord.

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. If the central angle is $\theta$, the inscribed angle is $\frac{\theta}{2}$.

Angles subtended by the same arc at any points on the remaining part of the circle are equal. This is a direct corollary of the angle at the centre theorem.

The angle subtended by a diameter at any point on the circumference is always a right angle ($90^\circ$). This is because the central angle is a straight line, which is $180^\circ$.

A quadrilateral is cyclic if all its vertices lie on a circle. 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 $\angle B + \angle D = 180^\circ$.

A quadrilateral is cyclic if the sum of any pair of its opposite angles is $180^\circ$.

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 are concyclic (lie on the same circle).

The unique circle passing through the three non-collinear vertices of a triangle is its circumcircle. Its centre, the circumcentre, is the point of intersection of the perpendicular bisectors of the sides.