Circles

A circle is a collection of all points 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.

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. Diameter divides a circle into two equal parts, each of which is called a semi circle.

Arc is a part of a circle. Minor arc is an arc of a circle whose length is less than that of a semi-circle. Major arc is an arc of a circle whose length is greater than that of a semi circle.

Sector is the region bounded by an arc of a circle and two radii. 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 (pi).

Properties

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.

There is one and only one circle passing through three non-collinear points.

Equal chords of a circle are equidistant from the centre, conversely chords that are equidistant from the centre of a circle are equal.

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.

Secants and Tangents

Secant is a line which intersects circle at two distinct points. Tangent is 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 or AB and EF of a circle intersect at a point P or Q outside or inside the circle, then PA × PB = PC × PD or QA × QB = QE × QF.

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.