tangent–secant theorem

{{short description|Geometry theorem relating line segments created by a secant and tangent line}}

File:Sekanten tangenten.svg,

$\backslash begin\{array\}\{cl\}$

\implies & \angle PG_2T = \angle PTG_1 \\[4pt]

\implies & \triangle PTG_2 \sim \triangle PG_1T \\[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 {{mvar|g}} intersecting the circle at points {{math|G₁}} and {{math|G₂}} and a tangent {{mvar|t}} intersecting the circle at point {{mvar|T}} and given that {{mvar|g}} and {{mvar|t}} intersect at point {{mvar|P}}, the following equation holds:

$$|PT|^2=|PG\_1|\backslash cdot|PG\_2|$$

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.