There is one and only one circle passing through three non-collinear points.
Two arcs of a circle are congruent if and only if the angles subtended by them at the centre are equal. Two arcs of a circle are congruent if and only if their corresponding chords are equal.
Equal chords of a circle subtend equal angles at the centre and conversely if the angles subtended by the chords at the centre of a circle are equal, then the chords are equal.
The perpendicular drawn from the centre of a circle to a chord bisects the chord. The line joining the centre of a circle to the mid point of a chord is perpendicular to the chord.
Equal chords of a circle are equidistant from the centre. Chords, that are equidistant from the centre of a circle, are equal.
The angle subtended at the centre of a circle by an arc is double the angle subtended by it on any point on the remaining part of the circle.
Angle in a semicircle is a right angle.
Angles in the same segment of a circle are equal. If a line segment joining two points subtends equal angles at two other points on the same side of the line containing the segment, the four points lie on a circle.