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.

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:

$$|PT|^2=|PG_1|\cdot|PG_2|$$

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.