A tangent to a circle is a line that touches the circle at exactly one point (point of tangency). Key theorem: The tangent to a circle is perpendicular to the radius at the point of tangency. From an external point, two tangents can be drawn to a circle, and they are equal in length.

Theorem 1: The tangent at any point of a circle is perpendicular to the radius through that point of contact. Theorem 2: The lengths of two tangents drawn from an external point to a circle are equal. These theorems are used to solve problems involving tangent lengths and angles.

No tangent can be drawn from a point inside the circle. Exactly one tangent can be drawn from a point on the circle. Exactly two tangents can be drawn from a point outside the circle. The angle between the two tangents drawn from an external point and the line joining the centre to that point is related by geometry.

The number of tangents possible depends on the position of the point relative to the circle: (1) Point inside the circle → 0 tangents; (2) Point on the circle → exactly 1 tangent (at that point); (3) Point outside the circle → exactly 2 tangents. If PA and PB are two tangents from external point P to a circle with centre O, then: PA = PB (tangent lengths are equal), OP bisects angle APB, OA ⊥ PA and OB ⊥ PB. The angle subtended at the centre by the two radii to tangent points = 180° − angle APB.