Intersecting secants theorem

Intersecting secants theorem {displaystyle triangle PACsim triangle PBD} Erträge {Anzeigestil |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 each other in P and some circle in A and D respective B and C the following equation holds: {Anzeigestil |PA|cdot |PD|=|PB|cdot |PC|} The theorem follows directly from the fact, that the triangles PAC und PBD are similar. They share {Anzeigestil 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.

Referenzen S. Gottwald: Die VNR Concise Encyclopedia of Mathematics. Springer, 2012, ISBN 9789401169820, pp. 175-176 Michael L.. O’Leary: Revolutionen in der Geometrie. 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 (Deutsch) External links Secant Secant Theorem at proofwiki.org Power of a Point Theorem auf cut-the-knot.org Weisstein, Erich W. ">

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