A circle is a collection of all points in a plane which are at a constant distance from a fixed point. The fixed point is called the centre of the circle.
A line segment joining centre of the circle to a point on the circle is called radius of the circle. The circle has infinite number of radii. All radii of a circle are equal.
A line segment joining any two points on the circle is called a chord. Chord passing through the centre of circle is called its diameter. Diameter is the longest chord of the circle.
Arc: A part of a circle.
Minor arc: An arc of a circle whose length is less than that of a semi-circle of the same circle.
Major arc: An arc of a circle whose length is greater than that of a semi circle of the same circle is called a major arc.
Diameter of a circle divides a circle into two equal arcs, each of which is called a semicircle.
Sector: The region bounded by an arc of a circle and two radii.
Segment: A chord divides the interior of a circle into two parts. Each of which is called a segment.
Circumference: The length of the boundary of a circle is the circumference of the circle. The ratio of the circumference of circle to its diameter is always a constant, which is denoted by Greek letter π.
Concyclic Points: Points which lie on a circle. Three non collinear points are always concyclic and a unique circle passes through them. There is one and only one circle passing through three non-collinear points.
Central Angle: Angle subtended by an arc at the centre of circle.
Arc, Chord, Central Angle
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. Conversely 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, conversely chords that are equidistant from the centre of a circle are equal.
Inscribed Angle: The angle subtended by an arc or chord on any point on the remaining part of circle.
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 semi circle is a right angle.
Cyclic Quadrilateral
A quadrilateral in which all four vertices lie on a circle.
If a pair of opposite angles of a quadrilateral is supplementary then the quadrilateral is cyclic.
If PQRS is a cyclic quadrilateral, ∠P + ∠R = 180° or ∠Q + ∠S = 180°.
If PQRS is a cyclic parallelogram then it is a rectangle.
Secants and Tangents
Secant: A line which intersects circle at two distinct points.
Tangent: A line which touches a circle at exactly one point and the point where it touches the circle is called point of contact.
When two points of intersection of secant and circle coincide it becomes a tangent.
- From an external point only two tangents can be drawn to a circle.
- The lengths of two tangents from an external point are equal.
- A radius through the point of contact is perpendicular to the tangent at the point.
- The tangents drawn from an external point to a circle are equally inclined to the line joining the point to the centre of circle.
- If two chords AB and CD of a circle intersect at a point P outside or inside the circle, then PA × PB = PC × PD.
- If PAB is a secant to a circle intersecting the circle at A and B and PT is a tangent to the circle at T, then PA × PB = PT2.
- The angles made by a chord in alternate segment through the point of contact of a tangent is equal to the angle between chord and tangent.