Tangent-secant theorem
From HandWiki
Short description: Relates line segments created by a secant with a tangent line
property of inscribed angles
[math]\displaystyle{ \Rightarrow }[/math]
[math]\displaystyle{ \angle PG_2T =\angle PTG_1 }[/math]
[math]\displaystyle{ \Rightarrow }[/math]
[math]\displaystyle{ \triangle PTG_2 \sim \triangle PG_1T }[/math]
[math]\displaystyle{ \Rightarrow }[/math]
[math]\displaystyle{ \frac{|PT|}{|PG_2|}=\frac{|PG_1|}{|PT|} }[/math]
[math]\displaystyle{ \Rightarrow }[/math]
[math]\displaystyle{ |PT|^2=|PG_1|\cdot|PG_2| }[/math]
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:
- [math]\displaystyle{ |PT|^2=|PG_1|\cdot|PG_2| }[/math]
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". http://mathworld.wolfram.com/Chord.html.
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