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Intersecting secants theorem

## Summary

The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.

${\displaystyle \triangle PAC\sim \triangle PBD}$
yields
${\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|}$

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:

${\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|}$

The theorem follows directly from the fact, that the triangles PAC and PBD are similar. They share ${\displaystyle \angle DPC}$ and ${\displaystyle \angle ADB=\angle ACB}$ 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:

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

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, 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)