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.

References

S. Gottwald: The VNR Concise Encyclopedia of Mathematics. Springer, 2012, ISBN 9789401169820, pp. 175-176
Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, ISBN 9780470591796, p. 161
Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, ISBN 978-3-411-04208-1, pp. 415-417 (German)

External links

Tangent Secant Theorem at proofwiki.org
Power of a Point Theorem auf cut-the-knot.org
Weisstein, Eric W. "Chord". MathWorld.

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This page was last edited on 30 August 2023, at 07:40

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