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:

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: 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 (a diameter) 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 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.

References

edit
  • 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)
edit

📚 Artikel Terkait di Wikipedia

Power of a point

Point Theorem at cut-the-knot Pythagorean Theorem (Proof #22) at cut-the-knot Intersecting Chords Theorem at cut-the-knot Intersecting Chords Theorem With

Intersecting secants theorem

geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the

Butterfly theorem

butterfly theorem is a classical result which can be stated as follows: Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and

Tangent–secant theorem

tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant

Geometric mean theorem

triangle. The theorem can also be thought of as a special case of the intersecting chords theorem for a circle, since the converse of Thales' theorem ensures

Circular arc

of the diameter, with length 2r − H. Applying the intersecting chords theorem to these two chords produces H ( 2 r − H ) = ( W 2 ) 2 , {\displaystyle

Ptolemy's theorem

CD}}\end{aligned}}} Casey's theorem Intersecting chords theorem Greek mathematics C. Ptolemy, Almagest, Book 1, Chapter 10. Wilson, Jim. "Ptolemy's Theorem." link verified

Inscribed angle

inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point