Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

s created by two intersecting

chords Chord or chords may refer to: Art and music * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord, a chord played on a guitar, which has a particular tuning * The Chords (British band), 1970s British mod ...

within a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

. 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 Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

's ''Elements''. More precisely, for two chords and intersecting in a point the following equation holds:

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

The converse is true as well. That is: If for two line segments and intersecting in the

equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...

above holds true, then their four endpoints lie on a common circle. Or in other words, if the

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...

s of a

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

intersect in and fulfill the equation above, then it is a

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

. The value of the two products in the chord theorem depends only on the distance of the intersection point from the circle's center and is called the ''

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

of the power of ''; more precisely, it can be stated that:

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

where is the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of the circle, and is the distance between the center of the circle and the intersection point . This property follows directly from applying the chord theorem to a third chord (a diameter) going through and the circle's center (see drawing). The theorem can be proven using

similar triangles In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly ...

(via the inscribed-angle theorem). Consider the angles of the triangles and :

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

This means the triangles and are similar and therefore

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

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 In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. Specifically, the power \Pi(P) of a point P with respect to ...

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