Tangent-secant theorem

The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle.

property of inscribed angles

${\displaystyle \Rightarrow }$

${\displaystyle \angle PG_{2}T=\angle PTG_{1}}$

${\displaystyle \Rightarrow }$

${\displaystyle \triangle PTG_{2}\sim \triangle PG_{1}T}$

${\displaystyle \Rightarrow }$

${\displaystyle {\frac {|PT|}{|PG_{2}|}}={\frac {|PG_{1}|}{|PT|}}}$

${\displaystyle \Rightarrow }$

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

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). Construct the diameter passing through point T, say it meets the circle at point D. Now, angles G₁DT and G₁G₂T are equal because they subtend the same arc of the circle. Triangle DG₁T is a right angled triangle because the inscribed angle DG₁T subtends a diameter DT. Since angle DTP is a right angle, angle PTG₁ is equal to G₁DT and hence equal to G₁G₂T. Therefore the triangles PG₁T and PTG₂ are similar, since two of their angles are equal, and thus corresponding sides of the triangles are proportional.

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.