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Wikipedia / 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 productss 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 and intersecting at point

Proof

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

By the inscribed angle theorem, and , as each pair intercepts a common chord. Opposite angles at the point are also congruent, . Therefore and are similar triangles, with congruent corresponding angles.

Related concepts

The converse is true as well. That is: If for two line segments and intersecting in the equation above holds true, then their four endpoints lie on a common circle. Or in other words, if the diagonals of a quadrilateral intersect in and fulfill the equation above, then it is a cyclic quadrilateral.

By applying the chord theorem to a third chord (a diameter) going through the circle's center and point where is the radius of the circle and is the distance between the circle's center and point .

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 from the circle's center to the intersection point is the absolute value of the power of 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, , p. 14

  • Hans Schupp: Elementargeometrie. Schöningh, Paderborn 1977, , p.149 (German).

  • Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, , pp.415-417 (German)