Tangent

Dependencies: Circle, Point–line axiom. A derived definition describing the limiting case in which a line meets a circle at exactly one point.

Definition

Let $\odot O$ be a circle, and let $\ell$ be a line.

If line $\ell$ and circle $\odot O$ have exactly one common point $T$, then $\ell$ is a tangent to the circle and $T$ is the point of tangency.

A line and a circle can have zero, one, or two common points. A tangent is the boundary case with exactly one: it touches the circle without crossing it.

Remarks

• External, tangent, or secant. Zero common points means external; one means tangent; two means secant, whose intersection points determine a chord. Equivalently, if $d$ is the distance from the center to the line and $r$ the radius, then $d>r$, $d=r$, and $d<r$ give the three cases.
• A tangent is perpendicular to the radius at the contact point. The tangent $\ell$ and radius $OT$ are perpendicular, the key starting fact in tangent problems.
• Tangents from an external point have equal length. The two tangent segments from one external point to a circle are equal (Two tangents from an external point have equal length).