Apollonius's theorem

Apollonius's theorem This article is about the lengths of the sides of a triangle. For his work on circles, see Problem of Apollonius. green/blue areas = red area Pythagoras as a special case: green area = red area In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".

Specifically, in any triangle {displaystyle ABC,} if {displaystyle AD} is a median, then {displaystyle |AB|^{2}+|AC|^{2}=2left(|AD|^{2}+|BD|^{2}right).} It is a special case of Stewart's theorem. For an isosceles triangle with {displaystyle |AB|=|AC|,} the median {displaystyle AD} is perpendicular to {displaystyle BC} and the theorem reduces to the Pythagorean theorem for triangle {displaystyle ADB} (or triangle {displaystyle ADC} ). From the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

The theorem is named for the ancient Greek mathematician Apollonius of Perga.

Contents 1 Proof 2 See also 3 References 4 External links Proof Proof of Apollonius's theorem The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines.[1] Let the triangle have sides {displaystyle a,b,c} with a median {displaystyle d} drawn to side {displaystyle a.} Let {displaystyle m} be the length of the segments of {displaystyle a} formed by the median, so {displaystyle m} is half of {displaystyle a.} Let the angles formed between {displaystyle a} and {displaystyle d} be {displaystyle theta } and {displaystyle theta ^{prime },} where {displaystyle theta } includes {displaystyle b} and {displaystyle theta ^{prime }} includes {displaystyle c.} Then {displaystyle theta ^{prime }} is the supplement of {displaystyle theta } and {displaystyle cos theta ^{prime }=-cos theta .} The law of cosines for {displaystyle theta } and {displaystyle theta ^{prime }} states that {displaystyle {begin{aligned}b^{2}&=m^{2}+d^{2}-2dmcos theta \c^{2}&=m^{2}+d^{2}-2dmcos theta '\&=m^{2}+d^{2}+2dmcos theta .,end{aligned}}} Add the first and third equations to obtain {displaystyle b^{2}+c^{2}=2(m^{2}+d^{2})} as required.

See also Formulas involving the medians' lengths – Line segment joining a triangle's vertex to the midpoint of the opposite side References ^ Godfrey, Charles; Siddons, Arthur Warry (1908). Modern Geometry. University Press. p. 20. External links Apollonius Theorem at PlanetMath. David B. Surowski: Advanced High-School Mathematics. p. 27 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: Euclidean geometryTheorems about triangles