## Circles - ESSENTIAL POINTS

• A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.
• The circle and its interior make up the circular region.
• A diameter is the longest chord and all diameters have the same length
• A piece of a circle between two points is called an arc.
• The region between a chord and either of its arcs is called a segment.
• The region between an arc and the two radii, joining the center to the end points of the arc is called a sector.
• Equal chords of a circle (or of congruent circles) subtend equal angles at the center.
• If the angles subtended by the chords of a circle (or of congruent circles) at the center (or centers) are equal, then the chords are equ
• The perpendicular drawn from the center of the circle to a chord bisects the chord.
• The line drawn through the center of a circle bisecting a chord is perpendicular to the chord.
• There is one and only one circle passing through three given non-collinear points.
• Equal chords of a circle (or of congruent circles) are equidistant from the center (or centers).
• Chords equidistant from the center of a circle are equal in length.
• If two chords of a circle are equal, then their corresponding arcs are congruent and vice-versa.
• Congruent arcs of a circle subtend equal angles at the center.
• The angle subtended by an arc at the center 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, then the four points are concyclic.
• The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.
• If the sum of a pair of opposite angles of a quadrilateral is 180 degrees, the quadrilateral is cyclic.

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