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{{short description|Geometry theorem relating line segments created by intersecting secants of a circle}}
[[File:Secant theorem.svg|thumb|upright=1.0|{{center|<math>\triangle PAC \sim \triangle PBD</math>}}{{center|yields}}{{center|<math>|PA|\cdot|PD|=|PB|\cdot|PC|</math>}}]]

In [[Euclidean geometry]], the '''intersecting secants theorem''' or just '''secant theorem''' describes the relation of [[line segment]]s created by two intersecting [[Secant line|secant]]s and the associated [[circle]].

For two lines {{mvar|AD}} and {{mvar|BC}} that [[Line–line intersection|intersect each other]] at {{mvar|P}} and for which {{math|''A'', ''B'', ''C'', ''D''}} all lie on the same circle, the following equation holds:

<math display=block>|PA|\cdot|PD| = |PB|\cdot|PC|</math>

The theorem follows directly from the fact that the [[triangle]]s {{math|△''PAC''}} and {{math|△''PBD''}} are [[Similarity (geometry)|similar]]. They share {{math|∠''DPC''}} and {{math|1=∠''ADB'' = ∠''ACB''}} as they are [[inscribed angle]]s over {{mvar|AB}}. The similarity yields an equation for [[ratio]]s which is equivalent to the equation of the theorem given above:
<math display=block>\frac{PA}{PC}=\frac{PB}{PD} \Leftrightarrow |PA|\cdot|PD|=|PB|\cdot|PC|</math>

Next to the [[intersecting chords theorem]] and the [[tangent-secant theorem]], the intersecting secants 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 point theorem]].

==References==
*S. Gottwald: ''The VNR Concise Encyclopedia of Mathematics''. Springer, 2012, {{ISBN|9789401169820}}, pp. [https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA175 175-176]
*Michael L. O'Leary: ''Revolutions in Geometry''. Wiley, 2010, {{ISBN|9780470591796}}, p. [https://books.google.com/books?id=Ch5CrMtyniEC&pg=PA161 161]
*''Schülerduden - Mathematik I''. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, {{ISBN|978-3-411-04208-1}}, pp. 415-417 (German)

== External links ==

* [https://proofwiki.org/wiki/Secant_Secant_Theorem ''Secant Secant Theorem''] at proofwiki.org
* [http://www.cut-the-knot.org/pythagoras/PPower.shtml ''Power of a Point Theorem''] auf cut-the-knot.org
* {{MathWorld|urlname=Chord|title=Chord}}

{{Ancient Greek mathematics}}

[[Category:Theorems about circles]]

Intersecting secants theorem - Revision history

imported>OlliverWithDoubleL: links, inline math formatting