Intersecting chords theorem

In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords 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 $A, B, C, D$ 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.

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 $△ BSC$ : ${\begin{aligned}\angle ADS&=\angle BCS\,({\text{inscribed angles over AB}})\\\angle DAS&=\angle CBS\,({\text{inscribed angles over CD}})\\\angle ASD&=\angle BSC\,({\text{opposing angles}})\end{aligned}}$ This means the triangles $△ ASD$ and $△ BSC$ are similar and therefore

${\frac {AS}{SD}}={\frac {BS}{SC}}\Leftrightarrow |AS|\cdot |SC|=|BS|\cdot |SD|$

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 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, ISBN 9789813100947, p. 14
Hans Schupp: Elementargeometrie. Schöningh, Paderborn 1977, ISBN 3-506-99189-2, p. 149 (German).
Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, ISBN 978-3-411-04208-1, pp. 415-417 (German)

External links

Intersecting Chords Theorem at cut-the-knot.org
Intersecting Chords Theorem Archived 2020-12-01 at the Wayback Machine at proofwiki.org
Weisstein, Eric W. "Chord". MathWorld.
Two interactive illustrations: [1] and [2]