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In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, a secant is a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
that intersects a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
at a minimum of two distinct
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
.. The word ''secant'' comes from the
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

Latin
word ''secare'', meaning ''to cut''. In the case of a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
, a secant intersects the circle at exactly two points. A
chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord (ast ...
is the
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

line segment
determined by the two points, that is, the interval on the secant whose ends are the two points.


Circles

A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a ''secant line'', at one point a ''tangent line'' and at no points an ''exterior line''. A ''chord'' is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of
plane geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a small ...
, results that seem obvious and were assumed (without statement) by
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
in his treatment, are usually proved. For example, ''Theorem (Elementary Circular Continuity)'': If \mathcal is a circle and \ell a line that contains a point that is inside \mathcal and a point that is outside of \mathcal then \ell is a secant line for \mathcal. In some situations phrasing results in terms of secant lines instead of chords can help to unify statements. As an example of this consider the result: :If two secant lines contain chords and in a circle and intersect at a point that is not on the circle, then the line segment lengths satisfy . If the point lies inside the circle this is Euclid III.35, but if the point is outside the circle the result is not contained in the Elements. However,
Robert Simson Robert Simson (14 October 1687 – 1 October 1768) was a Scotland, Scottish mathematics, mathematician and Professor of Mathematics, Glasgow, professor of mathematics at the University of Glasgow. The Simson line is named after him.
Robert Simson
following
Christopher Clavius Christopher Clavius (25 March 1538 – 6 February 1612) was a Jesuit The Society of Jesus (SJ; la, Societas Iesu) is a religious order of the Catholic Church The Catholic Church, often referred to as the Roman Catholic Church, ...

Christopher Clavius
demonstrated this result, sometimes called the
secant-secant theorem A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a pl ...
, in their commentaries on Euclid.


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
approximate An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very n ...
the
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
line to a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
, at some point , if it exists. Define a secant to a curve by two
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
, and , with fixed and variable. As approaches along the curve, if the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

slope
of the secant approaches a limit value, then that limit defines the slope of the tangent line at . The secant lines are the approximations to the tangent line. In calculus, this idea is the geometric definition of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

derivative
. A tangent line to a curve at a point may be a secant line to that curve if it intersects the curve in at least one point other than . Another way to look at this is to realize that being a tangent line at a point is a ''local'' property, depending only on the curve in the immediate neighborhood of , while being a secant line is a ''global'' property since the entire domain of the function producing the curve needs to be examined.


Sets and -secants

The concept of a secant line can be applied in a more general setting than Euclidean space. Let be a finite set of points in some geometric setting. A line will be called an -secant of if it contains exactly points of . For example, if is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant (or ''bisecant'') and a line passing through only one of them would be a 1-secant (or ''unisecant''). A unisecant in this example need not be a tangent line to the circle. This terminology is often used in
incidence geometry Incidence may refer to: Economics * Benefit incidence, the availability of a benefit * Expenditure incidence, the effect of government expenditure upon the distribution of private incomes * Fiscal incidence, the economic impact of government tax ...
and
discrete geometry Discrete geometry and combinatorial geometry are branches of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are re ...
. For instance, the
Sylvester–Gallai theorem The Sylvester–Gallai theorem in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concer ...
of incidence geometry states that if points of Euclidean geometry are not
collinear In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
then there must exist a 2-secant of them. And the original
orchard-planting problem In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. It is also called the tree-planting problem or simply the orchard p ...

orchard-planting problem
of discrete geometry asks for a bound on the number of 3-secants of a finite set of points. Finiteness of the set of points is not essential in this definition, as long as each line can intersect the set in only a finite number of points.


See also

*
Elliptic curve In , an elliptic curve is a , , of one, on which there is a specified point ''O''. An elliptic curve is defined over a ''K'' and describes points in ''K''2, the of ''K'' with itself. If the field's is different from 2 and 3, then the curv ...
, a curve for which every secant has a third point of intersection, from which most of a group law may be defined *
Mean value theorem In mathematics, the mean value theorem states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant line, secant through its endpoints. ...

Mean value theorem
, that every secant of the graph of a smooth function has a parallel tangent line *
Quadrisecant In geometry, a quadrisecant or quadrisecant line of a curve is a Line (geometry), line that passes through four points of the curve. Every knotted curve in three-dimensional Euclidean space has a quadrisecant. The number of quadrisecants of an algeb ...

Quadrisecant
, a line that intersects four points of a curve (usually a space curve) *
Secant plane Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproca ...
, the three-dimensional equivalent of a secant line * Secant variety, the union of secant lines and tangent lines to a given projective variety


References


External links

* {{Calculus topics Curves