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Geometry theorem relating line segments created by a secant and tangent line
Beginning with the
alternate segment theorem ,
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{\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}}}
In
Euclidean geometry , the tangent-secant theorem describes the relation of
line segments created by a
secant and a
tangent line with the associated
circle .
This result is found as Proposition 36 in Book 3 of
Euclid 's
Elements .
Given a secant g intersecting the circle at points G 1 and G 2 and a tangent t intersecting the circle at point T and given that g and t intersect at point P , the following equation holds:
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{\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|}
The tangent-secant theorem can be proven using similar triangles (see graphic).
Like the
intersecting chords theorem and the
intersecting 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, 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)
External links