Misplaced Pages

Tangent–secant theorem

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Geometry theorem relating line segments created by a secant and tangent line
Beginning with the alternate segment theorem, P G 2 T = P T G 1 P T G 2 P G 1 T | P T | | P G 2 | = | P G 1 | | P T | | P T | 2 = | P G 1 | | P G 2 | {\displaystyle {\begin{array}{cl}\implies &\angle PG_{2}T=\angle PTG_{1}\\\implies &\triangle PTG_{2}\sim \triangle PG_{1}T\\\implies &{\frac {|PT|}{|PG_{2}|}}={\frac {|PG_{1}|}{|PT|}}\\\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 g intersecting the circle at points G1 and G2 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 | | P G 2 | {\displaystyle |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

Ancient Greek mathematics
Mathematicians
(timeline)
Treatises
Problems
Concepts
and definitions
Results
In Elements
Apollonius
Other
Centers
Related
History of
Other cultures
Ancient Greece portal • icon Mathematics portal
Category: