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| A '''chord''' of a [[circle]] is a [[geometry|geometric]] [[line segment]] whose endpoints both lie on the circle. A [[secant line]], or just ''secant'', is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an [[Ellipse#Chords|ellipse]]. A chord that passes through a circle's center point is the circle's diameter.
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| [[Image:Chord in mathematics.svg|right|thumb|200px|The red segment ''BX'' is a '''chord''' <br />
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| (as is the diameter line ''AB'').]]
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| == Chords of a circle ==
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| {{Further2|[[Circle#Chord|Chord properties]]}}
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| Among properties of chords of a [[circle]] are the following:
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| # Chords are equidistant from the center only if their lengths are equal.
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| # A chord that passes through the center of a circle is called a diameter, and is the longest chord.
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| # 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]]).
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| The area that a circular chord "cuts off" is called a [[circular segment]].
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| Chord is from the Latin ''chorda'' meaning ''bowstring''.
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| ==Chords of an ellipse==
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| The midpoints of a set of parallel chords of an ellipse are [[Collinearity|collinear]].<ref>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.</ref>{{rp|p.147}}
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| == Chords in trigonometry ==
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| [[Image:TrigonometricChord.svg|left|200px]]
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| Chords were used extensively in the early development of [[trigonometry]]. The first known trigonometric table, compiled by [[Hipparchus]], [[table of chords|tabulated the value of the chord function]] for every 7.5 [[Degree (angle)|degree]]s. In the second century AD, [[Ptolemy]] of Alexandria compiled a more extensive table of chords in [[Almagest|his book on astronomy]], giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.
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| The chord function is defined geometrically as in the picture to the left. The chord of an [[angle]] is the [[length]] of the chord between two points on a unit circle separated by that angle. 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 (cos <math>\theta</math>, sin <math>\theta</math>), and then using the [[Pythagorean theorem]] to calculate the chord length:
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| : <math> \mathrm{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} =2 \sin \left(\frac{\theta }{2}\right). </math>
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| The last step uses the [[Trigonometric_identity#Double-angle.2C_triple-angle.2C_and_half-angle_formulae|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. The chord function satisfies many identities analogous to well-known modern ones:
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| {| class="wikitable"
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| !Name!!Sine-based!!Chord-based
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| |-
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| |Pythagorean
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| |<math>\sin^2 \theta + \cos^2 \theta = 1 \, </math>
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| |<math>\mathrm{crd}^2 \theta + \mathrm{crd}^2 (180^\circ - \theta) = 4 \, </math>
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| |-
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| |Half-angle
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| |<math>\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos \theta}{2}} \, </math>
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| |<math>\mathrm{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mathrm{crd}(180^\circ - \theta)} \,</math>
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| |-
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| | '''[[Apothem]] (a)'''
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| |<math>c=2 \sqrt{r^2- a^2}</math>
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| |<math>c=\sqrt{D ^2-4 a^2}</math>
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| |-
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| | '''Angle (θ)'''
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| |<math>c=2 r \sin \left(\frac{\theta }{2}\right)</math>
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| |<math>c=D \sin \left(\frac{\theta }{2}\right)</math>
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| |}
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| ==See also==
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| *[[Circular segment]]
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| *[[Circle graph]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| {{Commons category|Chord (geometry)}}
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| * [http://aleph0.clarku.edu/~djoyce/ma105/trighist.html History of Trigonometry Outline]
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| * [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html Trigonometric functions], focusing on history
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| *[http://www.mathopenref.com/chord.html Chord (of a circle)] With interactive animation
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| [[Category:Curves]]
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| [[Category:Trigonometry]]
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| [[ja:円 (数学)#円の性質]]
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