Circles

Circle is a set of points equidistant from fixed point called centre. The boundary of a circle is called its circumference. The distance between the circumference and the centre is called radius. A line segment that joins any two points on the circumference of the circle is called a chord of the circle. A chord of a circle passing through its centre is called the diameter of the circle. The portion (or part) of a circular region enclosed by two radii and the corresponding arc is called the sector of the circle. The portion (or part) of a circular region enclosed by a chord and the corresponding arc is called the segment of the circle.

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 (or 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 congruent circles) are equidistant from the centre. Equidistant chords from centre of a circle (or congruent circle) are equal. If two arcs of a circle (or congruent circle) are equal, corresponding chords are equal. Congruent arcs of a circle subtends equal angle at the centre. If two chords of a circle are equal, then their corresponding arcs are congruent and conversely, if two arcs are congruent, then their corresponding chords are equal. The angle subtended by an arc at the centre is double the angle subtended by it on the remaining part of the circle. Angles in the same segment of a circle are equal. Angle in a semicircle is 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 the circle. The angle subtended by an arc at the centre is double the angle subtended by it on the remaining part of the circle (i.e. they are concyclic). The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. If the sum of a pair of opposite angles of a quadrilateral is180º, the quadrilateral is cyclic.

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