Editing Secant line (section)

==Curves==
For curves more complicated than simple circles, the possibility that a line that intersects a curve in more than two distinct points arises. Some authors define a secant line to a curve as a line that intersects the curve in two distinct points. This definition leaves open the possibility that the line may have other points of intersection with the curve. When phrased this way the definitions of a secant line for circles and curves are identical and the possibility of additional points of intersection just does not occur for a circle.

===Secants and tangents===
Secants may be used to [[Approximation theory|approximate]] the [[tangent]] line to a [[curve]], at some point {{math|''P''}}, if it exists. Define a  secant to a curve by two [[Point (geometry)|points]], {{math|''P''}} and {{math|''Q''}}, with {{math|''P''}} fixed and {{math|''Q''}} variable. As {{math|''Q''}} approaches {{math|''P''}} along the curve, if the [[slope]] of the secant approaches a [[limit (mathematics)|limit value]], then that limit defines the slope of the tangent line at {{math|''P''}}.<ref name="cag"/> The secant lines {{math|{{overline|''PQ''}}}} are the approximations to the tangent line. In calculus, this idea is the geometric definition of the [[derivative]].
[[File:secanttangent.svg|thumb|The tangent line at point {{mvar|P}} is a secant line of the curve]]
A tangent line to a curve at a point {{math|''P''}} may be a secant line to that curve if it intersects the curve in at least one point other than {{math|''P''}}. Another way to look at this is to realize that being a tangent line at a point {{math|''P''}} is a ''local'' property, depending only on the curve in the immediate neighborhood of {{math|''P''}}, while being a secant line is a ''global'' property since the entire domain of the function producing the curve needs to be examined.