In case of intersection of a line and a circle, the following three possibilities are there:
A line which touches a circle at exactly one point is called a tangent line and the point where it touches the circle is called the point of contact.
A line which intersects the circle in two distinct points is called a secant.
A tangent is the limiting position of a secant when the two points of intersection coincide.
Tangent and Radius
A radius, though the point of contact of tangent to a circle, is perpendicular to the tangent at that point.
Tangent From Point Outside Circle
From an external point, two tangents can be drawn to a circle. The lengths of two tangents from an external point are equal.
PT = PT′
The tangents drawn from an external point to a circle are equally inclined to the line joining the point to the centre of the circle.
∠OPT = ∠OPT′
If two chords AB and CD of a circle intersect at a point P (inside or outside the circle), then
PA × PB = PC × PD
Intersecting Secant and Tangent
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
Angles Made by Tangent & Chord
The angles formed in the alternate segments by a chord through the point of contact of a tangent to a circle is equal to the angle between the chord and the tangent. This result is more commonly called as Angles in the Alternate Segment.
∠PRQ = ∠QPY
∠QPX = ∠QSP
If a line makes with a chord angles which are equal respectively to the angles formed by the chord in alternate segments, then the line is a tangent to the circle.