Secant
A secant is a line that intersects a circle at exactly two points, passing through the interior of the circle.
Formula
\text{secant-tangent: (external segment)} \times \text{(whole secant)} = (\text{tangent})^2
Definition
A secant is a line that crosses through a circle, entering at one point and exiting at another; the segment between those two points is a chord. The Secant-Tangent Theorem states that if a tangent and secant are drawn from the same external point, $(\text{tangent length})^2=(\text{external secant segment})\times(\text{whole secant length})$, and for two secants, $(\text{ext}_1)\times(\text{whole}_1)=(\text{ext}_2)\times(\text{whole}_2)$; both are instances of the power of a point, $\text{pow}(P,C)=|PO|^2-r^2$, positive outside the circle, zero on it, negative inside.
Example
Shining a laser through a ball would trace a secant, entering on one side and exiting the other; a diameter is a special chord of a secant that passes through the center. For an external point $P$ with tangent length $6$ and a secant entering the circle at $4$ units and exiting at $9$ units: $6^2=36=4\times9$, confirming the theorem. For $P$ outside a circle of radius $5$ with $|PO|=13$: power $=169-25=144$, tangent from $P=12$; a secant with near intersection $3$ from $P$ has far intersection $144/3=48$, and another with near intersection $4$ has far intersection $144/4=36$, all secants sharing the same power.
Key Insight
The difference between a secant and a tangent is that a secant crosses the circle at two points while a tangent just touches it at one, with the tangent as the limiting case when those two points merge. The power of a point theorem, that all secants from a fixed point share the same product, is a projective property following from the cross-ratio invariance of harmonic ranges on a circle, and the power function $\text{pow}(P,C)=|PO|^2-r^2$ defines a natural "distance" from a point to a circle used in radical axes and coaxial circle systems.