Intersecting chords theorem

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The intersecting chords theorem or just chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords in a circle. It states that the products of the lengths of the line segments on each chord are equal.

More precisely for two chords AC and BD intersecting in a point S the following equation holds:

${\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|}$

The converse is true as well, that is if for two line segments AC and BD intersecting in S the equation above holds, then their four endpoints A, B, C and 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:

${\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|=r^{2}-d^{2}}$

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:

${\displaystyle {\begin{aligned}\angle ADS&=\angle BCS\,({\text{inscribed angles over AB}})\\\angle DAS&=\angle CBS\,({\text{inscribed angles over CD}})\\\angle ASD&=\angle BSC\,({\text{opposing angles}})\end{aligned}}}$

This means the triangles ASD and BSC are similar and therefore

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

Next to the tangent-secant theorem and the intersecting secants theorem the intersecting chord 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.

Video Intersecting chords 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)

Maps Intersecting chords theorem

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]

Source of article : Wikipedia