A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.
Equal chords of a circle (or of congruent circles) subtend equal angles at the centre. If the angles subtended by two chords of a circle (or of congruent circles) at the centre (corresponding centres) are equal, the chords are equal.
The perpendicular from the centre of a circle to a chord bisects the chord. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
There is one and only one circle passing through three non-collinear points.
Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres). Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.
Congruent arcs of a circle subtend equal angles at the centre. 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.
Angles in the same segment of a circle are equal. Angle in a semicircle is a right angle. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.
The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. If sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.
Class 10
A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre).
A tangent to a circle is a line that intersects the circle at only one point.
The tangent to a circle is perpendicular to the radius through the point of contact.
The lengths of the two tangents from an external point to a circle are equal.