geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a secant is a line that intersects a

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

at a minimum of two distinct

points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...

.. The word ''secant'' comes from the

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

word ''secare'', meaning ''to cut''. In the case of a

circle 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. The distance between any point of the circle and the centre is cal ...

, a secant intersects the circle at exactly two points. A chord is the

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

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 ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, results that seem obvious and were assumed (without statement) by

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

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 Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him.Christopher Clavius demonstrated this result, sometimes called the intersecting secants theorem, 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 equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...

the

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

line to a

, at some point , if it exists. Define a secant to a curve by two

, and , with fixed and variable. As approaches along the curve, if the

slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...

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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

. 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 In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incide ...

and

discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geom ...

. For instance, the

Sylvester–Gallai theorem The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, ...

of incidence geometry states that if points of Euclidean geometry are not

collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

then there must exist a 2-secant of them. And the original 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.

References

External links