Secant
GeometryA 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. It cuts the circle in two points. A chord is the part of the secant that is inside the circle.
Example
If you shine a laser through a round ball (as if you could), the laser line is like a secant. It enters the circle on one side and exits the other. A diameter is a special chord of a secant passing through the center.
Key Insight
The difference between a secant and a tangent: a secant crosses the circle (two points), while a tangent just touches it (one point). A tangent is actually the limiting case of a secant when the two intersection points come together.
Definition
A secant is a line that intersects a circle at two distinct points. The segment between the two intersection points is a chord. Secant-Tangent Theorem: if a tangent and secant are drawn from the same external point, $(\text{tangent length})^2 = (\text{external secant segment}) \times (\text{whole secant length})$. Secant-Secant: $(\text{ext}_1) \times (\text{whole}_1) = (\text{ext}_2) \times (\text{whole}_2)$.
Example
External point $P$, tangent length $= 6$, secant through $P$ enters circle at $A$ (closer, $4$ units from $P$) and exits at $B$ (farther, $9$ units from $P$). Check: $6^2 = 36 = 4 \times 9 = 36$. The external segment $= 4$, whole secant $= 9$.
Key Insight
The secant-tangent theorem is a generalization of the power of a point: for any line through external point $P$ that intersects the circle, the product (near distance) $\times$ (far distance) is constant. This constant is the power of $P$, equaling $d^2 - r^2$ for a circle of radius $r$ and center-to-$P$ distance $d$.
Definition
A secant of a circle is a line (or line segment) meeting the circle in two points. In the power of a point framework: for a fixed point $P$ and circle $C$, the product $PA \times PB$ for any secant through $P$ intersecting $C$ at $A$ and $B$ equals the power $\text{pow}(P, C) = |PO|^2 - r^2$. This power is positive ($P$ outside), zero ($P$ on circle), or negative ($P$ inside). The secant generalizes to the concept of intersection multiplicity in algebraic geometry.
Example
$P$ outside circle ($r=5$, center $O$), $|PO|=13$: power $= 169-25=144$. Tangent from $P = 12$. Secant through $P$ with near intersection $3$ from $P$: far $= 144/3=48$. Verify: $3\times48=144$. Another secant near $4$: far$=144/4=36$. All secants from $P$ have the same power $144$.
Key Insight
The power of a point theorem - all secants from a fixed point have the same product - is a projective property. In projective geometry, this follows from the cross-ratio invariance of harmonic ranges on a circle. 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.