The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in ''A'' and ''D'' respective ''B'' and ''C'' the following equation holds: :

, PA, \cdot, PD, =, PB, \cdot, PC,

The theorem follows directly from the fact, that the triangles PAC and PBD are similar. They share

\angle DPC

and

\angle ADB=\angle ACB

as they are

inscribed angle In geometry, an 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 on the circle by two given points on the circle. Equivalently, an in ...

s over AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: :

\frac=\frac \Leftrightarrow , PA, \cdot, PD, =, PB, \cdot, PC,

Next to the

intersecting chords theorem The intersecting chords theorem or just the chord theorem is a statement in elementary geometry 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 o ...

and the

tangent-secant theorem The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's ''Elements''. Given a secant ''g'' intersectin ...

the intersecting secants 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

*S. Gottwald: ''The VNR Concise Encyclopedia of Mathematics''. Springer, 2012, , pp
175-176
*Michael L. O'Leary: ''Revolutions in Geometry''. Wiley, 2010, , p
161
*''Schülerduden - Mathematik I''. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, , pp. 415-417 (German)

External links

''Secant Secant Theorem''
at proofwiki.org
''Power of a Point Theorem''
auf cut-the-knot.org * {{Ancient Greek mathematics Theorems about circles