View source for Intersecting chords theorem

{{short description|Geometry theorem relating the line segments created by intersecting chords in a circle}}
[[File:Chord theorem.svg|thumb|upright=1.0|<math>|AS|\cdot|SC|=|BS|\cdot|SD|</math>]]
[[File:Chord theorem power.svg|thumb|upright=1.0|
<math>\begin{align}
& |AS|\cdot|SC| = |BS|\cdot|SD| \\ 
={}& (r+d)\cdot(r-d) = r^2-d^2
\end{align}</math>]]
[[File:Chord theorem proof.svg|thumb|upright=1.0|{{center|<math>\triangle ASD \sim \triangle BSC</math>}}]]

In [[Euclidean geometry]], the '''intersecting chords theorem''', or just the '''chord theorem''', is a statement that describes a relation of the four [[line segment]]s created by two intersecting [[Chord (geometry)|chords]] within a [[circle]]. 
It states that the [[Product (mathematics)|product]]s of the lengths of the line segments on each chord are equal. 
It is Proposition 35 of Book 3 of [[Euclid]]'s [[Euclid's Elements|''Elements'']]. 

More precisely, for two chords {{mvar|AC}} and {{mvar|BD}} intersecting in a point {{mvar|S}} the following equation holds:
<math display=block>|AS|\cdot|SC|=|BS|\cdot|SD|</math>

The converse is true as well. That is: If for two line segments {{mvar|{{overline|AC}}}} and {{mvar|{{overline|BD}}}} intersecting in {{mvar|S}} the [[equation]] above holds true, then their four endpoints {{math|''A'', ''B'', ''C'', ''D''}} lie on a common circle. Or in other words, if the [[diagonal]]s of a [[quadrilateral]] {{mvar|ABCD}} intersect in {{mvar|S}} and fulfill the equation above, then it is a [[cyclic quadrilateral]].

The value of the two products in the chord theorem depends only on the distance of the [[Line–line intersection|intersection point]] {{mvar|S}} from the circle's center and is called the ''[[absolute value]] of the [[power of a point|power of {{mvar|S}}]]''; more precisely, it can be stated that:
<math display=block>|AS|\cdot|SC| = |BS|\cdot|SD| = r^2-d^2,</math>
where {{mvar|r}} is the [[radius]] of the circle, and {{mvar|d}} is the distance between the center of the circle and the intersection point {{mvar|S}}. This property follows directly from applying the chord theorem to a third chord (a diameter) going through {{mvar|S}} and the circle's center {{mvar|M}} (see drawing).

The theorem can be proven using [[similar triangles]] (via the [[Inscribed angle|inscribed-angle theorem]]). Consider the angles of the triangles {{math|△''ASD''}} and {{math|△''BSC''}}:
<math display=block>\begin{align}
\angle ADS&=\angle BCS\, (\text{inscribed angles over AB})\\ 
\angle  DAS&=\angle CBS\, (\text{inscribed angles over CD})\\
\angle ASD&=\angle BSC\, (\text{opposing angles})
\end{align}</math>  
This means the triangles {{math|△''ASD''}} and {{math|△''BSC''}} are similar and therefore

<math display=block>\frac{AS}{SD}=\frac{BS}{SC} \Leftrightarrow |AS|\cdot|SC|=|BS|\cdot|SD|</math>

Next to the [[tangent-secant theorem]] and the [[intersecting secants theorem]], the intersecting chords theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the [[Power_of_a_point#Theorems|power of a point]] theorem.

==References==
*Paul Glaister: ''Intersecting Chords Theorem: 30 Years on''. Mathematics in School, Vol. 36, No. 1 (Jan., 2007), p. 22 ([https://www.jstor.org/stable/30215983 JSTOR])
*Bruce Shawyer: ''Explorations in Geometry''. World scientific, 2010, {{ISBN|9789813100947}}, p. [https://books.google.com/books?id=umNIDQAAQBAJ&pg=PA14 14]
* Hans Schupp: ''Elementargeometrie.'' Schöningh, Paderborn 1977, {{ISBN|3-506-99189-2}}, p.&nbsp;149 (German).
*''Schülerduden - Mathematik I''. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, {{ISBN|978-3-411-04208-1}}, pp.&nbsp;415-417 (German)

== External links ==
*[http://www.cut-the-knot.org/proofs/IntersectingChordsTheorem.shtml ''Intersecting Chords Theorem''] at cut-the-knot.org
*[http://www.proofwiki.org/wiki/Intersecting_Chord_Theorem ''Intersecting Chords Theorem''] {{Webarchive|url=https://web.archive.org/web/20201201083810/https://proofwiki.org/wiki/Intersecting_Chord_Theorem |date=2020-12-01 }} at proofwiki.org
*{{MathWorld|urlname=Chord|title=Chord}} 
*Two interactive illustrations: [https://www.geogebra.org/m/YwKdSERk] and [https://www.geogebra.org/m/kb4SK3YF]

{{Ancient Greek mathematics}}

[[Category:Theorems about circles]]