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A chord 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
is a
straight line segment Image:Segment definition.svg, 250px, The geometric definition of a closed line segment: the intersection (Euclidean geometry), intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry, a line s ...
whose endpoints both lie on a
circular arc Circular may refer to: * The shape of a circle * Circular (album), ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fal ...
. The
infinite line File:1D line.svg, 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. L ...
extension of a chord is a
secant line 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 ...
, or just ''secant''. More generally, a chord is a line segment joining two points on any curve, for instance, an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
. A chord that passes through a circle's center point is the circle's
diameter 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 ...

diameter
. The word ''chord'' is from the Latin ''chorda'' meaning ''bowstring''.


In circles

Among properties of chords 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
are the following: # Chords are equidistant from the center if and only if their lengths are equal. # Equal chords are subtended by equal angles from the center of the circle. # A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. # If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (
power of a point theorem In elementary plane 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 proper ...
).


In ellipses

The midpoints of a set of parallel chords of an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
are
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 ...
.


In trigonometry

Chords were used extensively in the early development of
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

trigonometry
. The first known trigonometric table, compiled by
Hipparchus Hipparchus of Nicaea (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the ...
, tabulated the value of the chord function for every
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
s. In the second century AD,
Ptolemy Claudius Ptolemy (; grc-koi, Κλαύδιος Πτολεμαῖος, , ; la, Claudius Ptolemaeus; AD) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes ...
of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from to 180 degrees by increments of degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part. The chord function is defined geometrically as shown in the picture. The chord of an
angle In Euclidean 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 c ...

angle
is the
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
of the chord between two points on a unit circle separated by that central angle. The angle ''θ'' is taken in the positive sense and must lie in the interval (radian measure). The chord function can be related to the modern
sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

sine
function, by taking one of the points to be (1,0), and the other point to be (), and then using the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

Pythagorean theorem
to calculate the chord length: : \operatorname\ \theta = \sqrt = \sqrt =2 \sin \left(\frac\right). The last step uses the
half-angle formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably a great deal was known about them. In the table below (where ''c'' is the chord length, and ''D'' the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones: The inverse function exists as well: :\theta = 2\arcsin\frac


See also

*
Circular segment 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 that ...

Circular segment
- the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary. *
Scale of chordsA scale of chords may be used to set or read an angle in the absence of a protractor. To draw an angle, compasses describe an arc from origin with a radius taken from the 60 mark. The required angle is copied from the scale by the compasses, and an a ...
*
Ptolemy's table of chordsThe table of chords, created by the Greece, Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's ''Almagest'', a treatise on mathematical astronom ...
*
Holditch's theorem In plane geometry, Holditch's theorem states that if a chord (geometry), chord of fixed length is allowed to rotate inside a convex closed curve, then the locus (mathematics), locus of a point on the chord a distance ''p'' from one end and a dista ...
, for a chord rotating in a convex closed curve *
Circle graph 175px, A circle with five chords and the corresponding circle graph. In graph theory, a circle graph is the intersection graph of a set of Chord (geometry), chords of a circle. That is, it is an undirected graph whose vertices can be associated w ...

Circle graph
*
Exsecant and excosecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, str ...
* Versine and haversine * Zindler curve (closed and simple curve in which all chords that divide the arc length into halves have the same length)


References


Further reading


External links


History of Trigonometry Outline


focusing on history

With interactive animation {{Authority control Curves Circles