Intersecting secants theorem

$\triangle PAC\sim \triangle PBD$

yields

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

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

${\frac {PA}{PC}}={\frac {PB}{PD}}\Leftrightarrow |PA|\cdot |PD|=|PB|\cdot |PC|$

References

• S. Gottwald: The VNR Concise Encyclopedia of Mathematics. Springer, 2012, ISBN 9789401169820, pp. 175-176
• Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, ISBN 9780470591796, p. 161
• Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, ISBN 978-3-411-04208-1, pp. 415-417 (German)

External links

• Secant Secant Theorem at proofwiki.org
• Power of a Point Theorem auf cut-the-knot.org
• Weisstein, Eric W. "Chord". MathWorld.