# Tangent-secant theorem

Tangent-secant theorem property of inscribed angles {displaystyle Rightarrow } {displaystyle angle PG_{2}T=angle PTG_{1}} {displaystyle Rightarrow } {displaystyle triangle PTG_{2}sim triangle PG_{1}T} {displaystyle Rightarrow } {displaystyle {frac {|PT|}{|PG_{2}|}}={frac {|PG_{1}|}{|PT|}}} {displaystyle Rightarrow } {displaystyle |PT|^{2}=|PG_{1}|cdot |PG_{2}|} 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: {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 Tangent Secant Theorem at proofwiki.org Power of a Point Theorem auf cut-the-knot.org Weisstein, Eric W. "Chord". MathWorld. hide vte Ancient Greek and Hellenistic mathematics (Euclidean geometry) Mathematicians (timeline) AnaxagorasAnthemiusArchytasAristaeus the ElderAristarchusApolloniusArchimedesAutolycusBionBrysonCallippusCarpusChrysippusCleomedesCononCtesibiusDemocritusDicaearchusDioclesDiophantusDinostratusDionysodorusDomninusEratosthenesEudemusEuclidEudoxusEutociusGeminusHeliodorusHeronHipparchusHippasusHippiasHippocratesHypatiaHypsiclesIsidore of MiletusLeonMarinusMenaechmusMenelausMetrodorusNicomachusNicomedesNicotelesOenopidesPappusPerseusPhilolausPhilonPhilonidesPorphyryPosidoniusProclusPtolemyPythagorasSerenus SimpliciusSosigenesSporusThalesTheaetetusTheanoTheodorusTheodosiusTheon of AlexandriaTheon of SmyrnaThymaridasXenocratesZeno of EleaZeno of SidonZenodorus Treatises AlmagestArchimedes PalimpsestArithmeticaConics (Apollonius)CatoptricsData (Euclid)Elements (Euclid)Measurement of a CircleOn Conoids and SpheroidsOn the Sizes and Distances (Aristarchus)On Sizes and Distances (Hipparchus)On the Moving Sphere (Autolycus)Euclid's OpticsOn SpiralsOn the Sphere and CylinderOstomachionPlanisphaeriumSphaericsThe Quadrature of the ParabolaThe Sand Reckoner Problems Constructible numbers Angle trisectionDoubling the cubeSquaring the circleProblem of Apollonius Concepts and definitions Angle CentralInscribedChordCircles of Apollonius Apollonian circlesApollonian gasketCircumscribed circleCommensurabilityDiophantine equationDoctrine of proportionalityGolden ratioGreek numeralsIncircle and excircles of a triangleMethod of exhaustionParallel postulatePlatonic solidLune of HippocratesQuadratrix of HippiasRegular polygonStraightedge and compass constructionTriangle center Results In Elements Angle bisector theoremExterior angle theoremEuclidean algorithmEuclid's theoremGeometric mean theoremGreek geometric algebraHinge theoremInscribed angle theoremIntercept theoremIntersecting chords theoremIntersecting secants theoremLaw of cosinesPons asinorumPythagorean theoremTangent-secant theoremThales's theoremTheorem of the gnomon Apollonius Apollonius's theorem Other Aristarchus's inequalityCrossbar theoremHeron's formulaIrrational numbersLaw of sinesMenelaus's theoremPappus's area theoremProblem II.8 of ArithmeticaPtolemy's inequalityPtolemy's table of chordsPtolemy's theoremSpiral of Theodorus Centers CyreneLibrary of AlexandriaPlatonic Academy Other Ancient Greek astronomyGreek numeralsLatin translations of the 12th centuryNeusis construction  Ancient Greece portal •  Mathematics portal Categories: Theorems about circles

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