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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:
:
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.
175-176
*Michael L. O'Leary: ''Revolutions in Geometry''. Wiley, 2010, , p
161
*''Schülerduden - Mathematik I''. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, , pp. 415-417 (German)
''Tangent Secant Theorem''
at proofwiki.org
''Power of a Point Theorem''
auf cut-the-knot.org * {{Ancient Greek mathematics Theorems about circles
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:
:
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, , pp175-176
*Michael L. O'Leary: ''Revolutions in Geometry''. Wiley, 2010, , p
161
*''Schülerduden - Mathematik I''. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, , pp. 415-417 (German)
External links
''Tangent Secant Theorem''
at proofwiki.org
''Power of a Point Theorem''
auf cut-the-knot.org * {{Ancient Greek mathematics Theorems about circles