A circle is the set of points in a plane at a fixed distance r from a fixed centre O. Every diameter is an axis of reflection symmetry, and rotation about O through any angle maps the circle onto itself. A radius joins O to a point on the circle; a chord joins two points on it.

There are infinitely many circles through two distinct points: each possible centre lies on the perpendicular bisector of the segment joining them. Three non-collinear points determine exactly one circle. Its centre, the circumcentre, is where the perpendicular bisectors of the triangle’s sides meet. Three collinear points do not determine a circle.

Equal chords of the same circle subtend equal angles at its centre, and equal central angles subtend equal chords. A perpendicular drawn from the centre to a chord bisects that chord; conversely, the line from the centre to the midpoint of a chord is perpendicular to it. These results connect the chord’s length with its position in the circle.

Chords of equal length are equally distant from the centre. Conversely, chords at equal perpendicular distances from the centre have equal lengths. Of two unequal chords in the same circle, the longer one is nearer to the centre. Distance here means the perpendicular distance from the centre to the chord.

An arc subtends an angle at the centre twice as large as the angle it subtends at a point on the remaining circumference. Therefore, an angle subtended by a diameter is 90°. A quadrilateral whose four vertices lie on one circle is cyclic; its opposite angles sum to 180°. Conversely, if one pair of opposite angles of a quadrilateral sums to 180°, the quadrilateral is cyclic.