Intersecting secants theorem

File:Secant theorem.svg

$\mathrm{△}PAC\sim \mathrm{△}PBD$

yields

$|PA|\cdot |PD|=|PB|\cdot |PC|$

In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.

For two lines Template:Mvar and Template:Mvar that intersect each other at Template:Mvar and for which A, B, C, D all lie on the same circle, 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 ∠DPC and ∠ADB = ∠ACB as they are inscribed angles over Template:Mvar. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: $$\frac{PA}{PC}=\frac{PB}{PD}\iff |PA|\cdot |PD|=|PB|\cdot |PC|$$

Next to the intersecting chords theorem and the tangent-secant theorem, 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, Template:ISBN, pp. 175-176
• Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, Template:ISBN, p. 161
• Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, Template:ISBN, pp. 415-417 (German)

External links

• Secant Secant Theorem at proofwiki.org
• Power of a Point Theorem auf cut-the-knot.org