Tangent–secant theorem

Beginning with the alternate segment theorem, $\begin{array}{cc}⟹& \mathrm{\angle }P{G}_{2}T=\mathrm{\angle }PT{G}_1\\ ⟹& \mathrm{△}PT{G}_2\sim \mathrm{△}P{G}_{1}T\\ ⟹& \frac{|PT|}{|P{G}_2|}=\frac{|P{G}_1|}{|PT|}\\ ⟹& |PT{|}^2=|P{G}_1|\cdot |P{G}_2|\end{array}$

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 Template:Mvar intersecting the circle at points Template:Math and Template:Math and a tangent Template:Mvar intersecting the circle at point Template:Mvar and given that Template:Mvar and Template:Mvar intersect at point Template:Mvar, the following equation holds:

$$|PT{|}^2=|P{G}_1|\cdot |P{G}_2|$$

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, Template:ISBN, pp. 175-176
• Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, Template:ISBN, p. 161
• Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, Template:ISBN, pp. 415-417 (German)

External links

• Tangent Secant Theorem at proofwiki.org
• Power of a Point Theorem auf cut-the-knot.org
• Template:MathWorld