Geometry theorem relating line segments created by a secant and tangent line
Beginning with thealternate segment theorem ,⟹ ∠ P G 2 T = ∠ P T G 1 ⟹ △ P T G 2 ∼ △ P G 1 T ⟹ | P T | | P G 2 | = | P G 1 | | P T | ⟹ | P T | 2 = | P G 1 | ⋅ | P G 2 | {\displaystyle {\begin{array}{cl}\implies &\angle PG_{2}T=\angle PTG_{1}\\[4pt]\implies &\triangle PTG_{2}\sim \triangle PG_{1}T\\[4pt]\implies &{\frac {|PT|}{|PG_{2}|}}={\frac {|PG_{1}|}{|PT|}}\\[2pt]\implies &|PT|^{2}=|PG_{1}|\cdot |PG_{2}|\end{array}}} InEuclidean geometry , thetangent-secant theorem describes the relation ofline segments created by asecant and atangent line with the associatedcircle . This result is found as Proposition 36 in Book 3 ofEuclid 'sElements
Given a secantg intersecting the circle at pointsG 1 G 2 t intersecting the circle at pointT and given thatg andt intersect at pointP , the following equation holds:
| P T | 2 = | P G 1 | ⋅ | P G 2 | {\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|}
The tangent-secant theorem can be proven using similar triangles (see graphic).
Like theintersecting chords theorem and theintersecting secants theorem , the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, thepower of point theorem .
S. Gottwald:The VNR Concise Encyclopedia of Mathematics . Springer, 2012,ISBN 9789401169820 175-176 Michael L. O'Leary:Revolutions in Geometry . Wiley, 2010,ISBN 9780470591796 161 Schülerduden - Mathematik I . Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008,ISBN 978-3-411-04208-1
