Tangent–secant theorem

Beginning with the alternate segment theorem,

\begin{matrix} ⟹ & ∠ P G_{2} T = ∠ P T G_{1} \\ ⟹ & △ P T G_{2} \sim △ P G_{1} T \\ ⟹ & \frac{| P T |}{| P G_{2} |} = \frac{| P G_{1} |}{| P T |} \\ ⟹ & | P T |^{2} = | P G_{1} | \cdot | P G_{2} | \end{matrix}

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:

$| P T |^{2} = | P G_{1} | \cdot | P G_{2} |$

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, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)., pp. 175-176
Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)., p. 161
Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)., pp. 415-417 (German)

External links

Tangent Secant Theorem at proofwiki.org
Power of a Point Theorem auf cut-the-knot.org
Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

Tangent–secant theorem

References

External links

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