Tangent–secant theorem

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.

Beginning with the alternate segment theorem, ${\displaystyle {\begin{array}{cl}\implies &\angle PG_{2}T=\angle PTG_{1}\\[4pt]\implies &\triangle PTG_{2}\sim \triangle PG_{1}T\\[4pt]\implies &{\frac {|PT|}{|PG_{2}|}}={\frac {|PG_{1}|}{|PT|}}\\[2pt]\implies &|PT|^{2}=|PG_{1}|\cdot |PG_{2}|\end{array}}}$

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.