Key Points

A tangent to a circle is a line that intersects the circle at exactly one point. This unique point is called the point of contact.

A secant is a line that intersects a circle at two distinct points. A tangent can be considered a special case of a secant where the two intersection points coincide.

Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact. If a tangent touches a circle with centre $O$ at point $P$, then the radius $OP$ is perpendicular to the tangent, meaning the angle formed is $90^\circ$.

From a point inside a circle, zero tangents can be drawn. From a point on the circle, exactly one tangent can be drawn. From a point outside the circle, exactly two tangents can be drawn.

Theorem: The lengths of the two tangents drawn from an external point to a circle are equal. If $P$ is an external point and $PA$ and $PB$ are tangents with points of contact $A$ and $B$, then $PA = PB$.

In problems involving tangents, a right-angled triangle is formed by the radius, the tangent, and the line from the centre to the external point. For a tangent $PT$ from point $P$ to a circle with centre $O$ and radius $r=OT$, we have $OP^2 = OT^2 + PT^2$.

The line segment connecting the centre of the circle to an external point bisects the angle between the two tangents drawn from that point. In a circle with centre $O$, for tangents $PA$ and $PB$, the line $OP$ is the angle bisector of $\angle APB$.

The angle between the two tangents from an external point is supplementary to the angle subtended by the points of contact at the centre. If $P$ is the external point and $A, B$ are points of contact, then $\angle APB + \angle AOB = 180^\circ$.

The tangents drawn at the two endpoints of a diameter of a circle are always parallel to each other.

The perpendicular line drawn at the point of contact to a tangent of a circle always passes through the centre of the circle.

If a quadrilateral $ABCD$ is drawn to circumscribe a circle, the sums of its opposite sides are equal. This means $AB + CD = AD + BC$.

A parallelogram that circumscribes a circle must be a rhombus. This is a direct consequence of the property that sums of opposite sides are equal.

For two concentric circles, a chord of the larger circle that is a tangent to the smaller circle is bisected at the point of contact.

The opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. For quadrilateral $ABCD$ with centre $O$, $\angle AOB + \angle COD = 180^\circ$ and $\angle BOC + \angle DOA = 180^\circ$.