Tangent–secant theorem

Geometry theorem relating line segments created by a secant and tangent line

Beginning with the alternate segment theorem,

{\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}}

{\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 G₁ and G₂ and a tangent t intersecting the circle at point T and given that g and t intersect at point P, the following equation holds:

$|PT|^{2}=|PG_{1}|\cdot |PG_{2}|$ {\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

[edit ]

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

[edit ]

Tangent Secant Theorem at proofwiki.org
Power of a Point Theorem auf cut-the-knot.org
Weisstein, Eric W. "Chord". MathWorld .

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