${\displaystyle \Rightarrow }$
${\displaystyle \angle PG_{2}T=\angle PTG_{1))$
${\displaystyle \Rightarrow }$
${\displaystyle \triangle PTG_{2}\sim \triangle PG_{1}T}$
${\displaystyle \Rightarrow }$
${\displaystyle {\frac {|PT|}{|PG_{2}|))={\frac {|PG_{1}|}{|PT|))}$
${\displaystyle \Rightarrow }$
${\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|}$

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:

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

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)