Intersecting chords theorem

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

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.

For example, for two chords Template:Mvar and Template:Mvar intersecting at point Template:Mvar, the following equation holds: $$|AS|\cdot |SC|=|BS|\cdot |SD|$$

Proof

The theorem can be proven by triangle similarity and the inscribed angle theorem.

Template:Center

By the inscribed angle theorem, Template:Tmath and Template:Tmath, as each pair intercepts a common chord. Opposite angles at the point Template:Tmath are also congruent, Template:Tmath. Therefore Template:Tmath and Template:Tmath are similar triangles, with congruent corresponding angles.

The sides of one triangle are uniformly proportional to the corresponding sides of any similar triangle, so $$\frac{|AS|}{|BS|}=\frac{|SD|}{|SC|},$$ or, equivalently, $$|AS|\cdot |SC|=|BS|\cdot |SD|.$$

Related concepts

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 Template:Math 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.

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

By applying the chord theorem to a third chord (a diameter) going through the circle's center Template:Mvar and point Template:Mvar, the following relationships can be made: $$|AS|\cdot |SC|=|BS|\cdot |SD|=r^2-d^2,$$ where Template:Mvar is the radius of the circle and Template:Mvar is the distance between the circle's center Template:Mvar and point Template:Mvar.

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. The distance Template:Mvar from the circle's center Template:Mvar to the intersection point Template:Mvar is the absolute value of the power of Template:Mvar with respect to the circle.

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
• Template:MathWorld
• Two interactive illustrations: [1] and [2]