The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting

chords Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord (as ...

within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's ''Elements''. More precisely, for two chords ''AC'' and ''BD'' intersecting in a point ''S'' the following equation holds: :

, AS, \cdot, SC, =, BS, \cdot, SD,

The converse is true as well, that is if for two line segments ''AC'' and ''BD'' intersecting in S the equation above holds true, then their four endpoints ''A'', ''B'', ''C'' and ''D'' lie on a common circle. Or in other words if the diagonals of a quadrilateral ''ABCD'' intersect in ''S'' and fulfill the equation above then it is a

cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...

. The value of the two products in the chord theorem depends only on the distance of the intersection point ''S'' from the circle's center and is called the absolute value of the power of ''S'', more precisely it can be stated that: :

, AS, \cdot, SC, =, BS, \cdot, SD, =r^2-d^2

where ''r'' is the radius of the circle, and ''d'' is the distance between the center of the circle and the intersection point ''S''. This property follows directly from applying the chord theorem to a third chord going through ''S'' and the circle's center ''M'' (see drawing). The theorem can be proven using similar triangles (via the inscribed-angle theorem). Consider the angles of the triangles ''ASD'' and ''BSC'': :

\begin
\angle ADS&=\angle BCS\, (\text)\\ 
\angle  DAS&=\angle CBS\, (\text)\\
\angle ASD&=\angle BSC\, (\text)
\end

This means the triangles ''ASD'' and ''BSC'' are similar and therefore :

\frac=\frac \Leftrightarrow , AS, \cdot, SC, =, BS, \cdot, SD,

Next to the

tangent-secant theorem The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's ''Elements''. Given a secant ''g'' intersectin ...

and the

intersecting secants theorem The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in '' ...

the intersecting chord theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem.

References

*Paul Glaister: ''Intersecting Chords Theorem: 30 Years on''. Mathematics in School, Vol. 36, No. 1 (Jan., 2007), p. 22
JSTOR
*Bruce Shawyer: ''Explorations in Geometry''. World scientific, 2010, , p
14
* Hans Schupp: ''Elementargeometrie.'' Schöningh, Paderborn 1977, , p. 149 (German). *''Schülerduden - Mathematik I''. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, , pp. 415-417 (German)

External links

''Intersecting Chords Theorem''
at cut-the-knot.org
''Intersecting Chords Theorem''
at proofwiki.org * *Two interactive illustrations

an

{{Ancient Greek mathematics Theorems about circles