Intersecting chords theorem

Geometry theorem relating the line segments created by intersecting chords in a circle
| A S | ⋅ | S C | = | B S | ⋅ | S D | {\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|}
| A S | ⋅ | S C | = | B S | ⋅ | S D | = ( r + d ) ⋅ ( r − d ) = r 2 − d 2 {\displaystyle {\begin{aligned}&|AS|\cdot |SC|=|BS|\cdot |SD|\\={}&(r+d)\cdot (r-d)=r^{2}-d^{2}\end{aligned}}}
△ A S D ∼ △ B S C {\displaystyle \triangle ASD\sim \triangle 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 AC and BD intersecting in a point S the following equation holds:

| A S | ⋅ | S C | = | B S | ⋅ | S D | {\displaystyle |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:

| A S | ⋅ | S C | = | B S | ⋅ | S D | = r 2 − d 2 {\displaystyle |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:

∠ A D S = ∠ B C S ( inscribed angles over AB ) ∠ D A S = ∠ C B S ( inscribed angles over CD ) ∠ A S D = ∠ B S C ( opposing angles ) {\displaystyle {\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

A S S D = B S S C ⇔ | A S | ⋅ | S C | = | B S | ⋅ | S D | {\displaystyle {\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 at proofwiki.org
  • Weisstein, Eric W. "Chord". MathWorld.
  • Two interactive illustrations: [1] and [2]
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