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 AC and BD intersecting at point S, the following equation holds: $$|AS|\cdot |SC|=|BS|\cdot |SD|$$

Proof

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

$\triangle ASD\sim \triangle BSC$

By the inscribed angle theorem, ⁠$\angle BCA=\angle ADB$⁠ and ⁠$\angle DAC=\angle CBD$⁠, as each pair intercepts a common chord. Opposite angles at the point ⁠$S$⁠ are also congruent, ⁠$\angle ASD=\angle BSC$⁠. Therefore ⁠$\triangle ASD$⁠ and ⁠$\triangle BSC$⁠ 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 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.

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

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

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 d from the circle's center M to the intersection point S is the absolute value of the power of S 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, 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: and