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.
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For two lines AD and BC that intersect each other at P and for which A, B, C, D all lie on the same circle, the following equation holds:
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 AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above:
intersecting, secants, theorem, euclidean, geometry, intersecting, secants, theorem, just, secant, theorem, describes, relation, line, segments, created, intersecting, secants, associated, circle, displaystyle, triangle, triangle, yields, displaystyle, cdot, c. 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 P A C P B D displaystyle triangle PAC sim triangle PBD yields P A P D P B P C displaystyle PA cdot PD PB cdot PC For two lines AD and BC that intersect each other at P and for which A B C D all lie on the same circle the following equation holds P A P D P B P C displaystyle 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 AB The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above P A P C P B P D P A P D P B P C 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 editS 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 Schulerduden Mathematik I Bibliographisches Institut amp F A Brockhaus 8 Auflage Mannheim 2008 ISBN 978 3 411 04208 1 pp 415 417 German External links editSecant Secant Theorem at proofwiki org Power of a Point Theorem auf cut the knot org Weisstein Eric W Chord MathWorld Retrieved from https en wikipedia org w index php title Intersecting secants theorem amp oldid 1172993726, wikipedia, wiki, book, books, library,
