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A chord of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
is a straight line segment whose endpoints both lie on a circular arc. The infinite line extension of a chord is a secant line, or just ''secant''. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
. The word ''chord'' is from the Latin ''chorda'' meaning ''
bowstring A bowstring joins the two ends of the bow stave and launches the arrow. Desirable properties include light weight, strength, resistance to abrasion, and resistance to water. Mass has most effect at the center of the string; of extra mass in th ...
''.


In circles

Among properties of chords of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
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 conics

The midpoints of a set of parallel chords of a conic are collinear ( midpoint theorem for conics).


In trigonometry

Chords were used extensively in the early development of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
. The first known trigonometric table, compiled by
Hipparchus Hipparchus (; 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 the precession of the e ...
, tabulated the value of the chord function for every degrees. In the second century AD,
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
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, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
is the
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
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 function, by taking one of the points to be (1,0), and the other point to be (), and then using the Pythagorean theorem to calculate the chord length: : \operatorname\ \theta = \sqrt = \sqrt =2 \sin \left(\frac\right). The last step uses the half-angle formula. 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, a circular segment (symbol: ), also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is ...
- 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 chords * Ptolemy's table of chords * Holditch's theorem, for a chord rotating in a convex closed curve *
Circle graph In graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the ...
* Exsecant and excosecant * 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 Circles Curves Geometry