Circle

An arc whose length is less than that of a semicircle is called a minor arc.

An arc whose length is greater than that of a semicircle is called a major arc.

A line which intersects a circle at two points is called a secant line.

A line which intersects(touches) a circle at only one point is called a tangent line.

The perpendicular from the centre of a circle to a chord bisects the chord. (refer fig1)

The line segment joining the centre of a circle to the midpoint of a chord is perpendicular to the chord. (refer fig1)

In fig 1,

AB is a chord, C is the centre, CP is perpendicular to AB, AP = PB

Equal chords of a circle are equidistant from the centre of the circle. (Converse is also true)

If the angles subtended at the centre by two chords of a circle are equal, then the chords are equal in length. (Converse is also true)

If fig 2 , angle ACB is twice angle APB where C is the centre of the circle.

Angles in the same segment of a circle are equal (refer fig.3)

angle APB and angle AQB are equal. (C is the centre)

Angle in a semicircle is a right angle.

Opposite angles of a cyclic quadrilateral are supplementary.

The tangent at any point of a circle is perpendicular to the diameter (radius) through the point of contact.

The length of two tangents drawn from an external point to a particular circle are equal in length.

If two chords AB and PQ of a circle intersect at R then AR.BR = PR.RQ

(refer following figures)

Let PAB be a secant and PT be a tangent segment the PA.PB = PT ²

(refer following figure)

If two circles touch (internally or externally), then the point of contact lies on the line joining the centres of the two circles.

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