Intersecting chords theorem

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:

|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:

|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 $△ BSC$ :

{\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

{\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.

YouTube Encyclopedic

1/5
Views:
10 270
13 126
365 735
2 432 399
9 728

Transcription

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]

Ancient Greek mathematics

Mathematicians
(timeline)

Treatises

Problems

Concepts
and definitions

Results

In Elements	Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon
Apollonius	Apollonius's theorem
Other	Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Law of sines Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus

Centers

Ancient Greek astronomy Attic numerals Greek numerals Latin translations of the 12th century Non-Euclidean geometry Philosophy of mathematics Neusis construction
History of	A History of Greek Mathematics by Thomas Heath algebra timeline arithmetic timeline calculus timeline geometry timeline logic timeline mathematics timeline numbers prehistoric counting numeral systems list
Other cultures	Arabian/Islamic Babylonian Chinese Egyptian Incan Indian Japanese

Ancient Greece portal •

Mathematics portal

This page was last edited on 20 December 2023, at 16:07

From Wikipedia, the free encyclopedia

YouTube Encyclopedic

Transcription

References

External links