Intersecting chords theorem

File:Chord theorem.svg

$|AS|\cdot |SC|=|BS|\cdot |SD|$

File:Chord theorem power.svg

$\begin{array}{rl}& |AS|\cdot |SC|=|BS|\cdot |SD|\\ =& (r+d)\cdot (r-d)=r^2-d^2\end{array}$

File:Chord theorem proof.svg

$\mathrm{△}ASD\sim \mathrm{△}BSC$

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 Template:Mvar and Template:Mvar intersecting in a point Template:Mvar the following equation holds: $$|AS|\cdot |SC|=|BS|\cdot |SD|$$

The converse is true as well. That is: If for two line segments Template:Mvar and Template:Mvar intersecting in Template:Mvar 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 Template:Mvar intersect in Template:Mvar 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 Template:Mvar from the circle's center and is called the absolute value of the [[power of a point|power of Template:Mvar]]; more precisely, it can be stated that: $$|AS|\cdot |SC|=|BS|\cdot |SD|=r^2-d^2,$$ where Template:Mvar is the radius of the circle, and Template:Mvar is the distance between the center of the circle and the intersection point Template:Mvar. This property follows directly from applying the chord theorem to a third chord (a diameter) going through Template:Mvar and the circle's center Template:Mvar (see drawing).

The theorem can be proven using similar triangles (via the inscribed-angle theorem). Consider the angles of the triangles △ASD and △BSC: $$\begin{array}{rl}\mathrm{\angle }ADS& =\mathrm{\angle }BCS(\text{inscribed angles over AB})\\ \mathrm{\angle }DAS& =\mathrm{\angle }CBS(\text{inscribed angles over CD})\\ \mathrm{\angle }ASD& =\mathrm{\angle }BSC(\text{opposing angles})\end{array}$$ This means the triangles △ASD and △BSC are similar and therefore

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

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 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, Template:ISBN, p. 14
• Hans Schupp: Elementargeometrie. Schöningh, Paderborn 1977, Template:ISBN, p. 149 (German).
• Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, Template:ISBN, pp. 415-417 (German)

External links

• Intersecting Chords Theorem at cut-the-knot.org
• Intersecting Chords Theorem Template:Webarchive at proofwiki.org
• Two interactive illustrations: [1] and [2]