“Intersecting chords theorem”的意思、由来-开放百科全书

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Intersecting chords theorem

释义

References
External links

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:

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:

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:

This means the triangles ASD and BSC are similar and therefore

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.

References

Paul Glaister: Intersecting Chords Theorem: 30 Years on. Mathematics in School, Vol. 36, No. 1 (Jan., 2007), p. 22 ([https://www.jstor.org/stable/30215983 JSTOR])
Bruce Shawyer: Explorations in Geometry. World scientific, 2010, {{ISBN|9789813100947}}, p. [https://books.google.de/books?id=umNIDQAAQBAJ&pg=PA14 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
{{MathWorld|urlname=Chord|title=Chord}}
Two interactive illustrations: [https://www.geogebra.org/m/YwKdSERk] and [https://www.geogebra.org/m/kb4SK3YF]

1 : Theorems in plane geometry

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