Pappus's area theorem

Pappus's area theorem dark grey area = light grey area Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria (4th century AD), who discovered it.

Contents 1 Theorem 2 Proof 3 References 4 External links Theorem Given an arbitrary triangle with two arbitrary parallelograms attached to two of its sides the theorem tells how to construct a parallelogram over the third side, such that the area of the third parallelogram equals the sum of the areas of the other two parallelograms.

Let ABC be the arbitrary triangle and ABDE and ACFG the two arbitrary parallelograms attached to the triangle sides AB and AC. The extended parallelogram sides DE and FG intersect at H. The line segment AH now "becomes" the side of the third parallelogram BCML attached to the triangle side BC, i.e., one constructs line segments BL and CM over BC, such that BL and CM are a parallel and equal in length to AH. The following identity then holds for the areas (denoted by A) of the parallelograms: {displaystyle {text{A}}_{ABDE}+{text{A}}_{ACFG}={text{A}}_{BCML}} The theorem generalizes the Pythagorean theorem twofold. Firstly it works for arbitrary triangles rather than only for right angled ones and secondly it uses parallelograms rather than squares. For squares on two sides of an arbitrary triangle it yields a parallelogram of equal area over the third side and if the two sides are the legs of a right angle the parallelogram over the third side will be square as well. For a right-angled triangle, two parallelograms attached to the legs of the right angle yield a rectangle of equal area on the third side and again if the two parallelograms are squares then the rectangle on the third side will be a square as well.

Proof Due to having the same base length and height the parallelograms ABDE and ABUH have the same area, the same argument applying to the parallelograms ACFG and ACVH, ABUH and BLQR, ACVH and RCMQ. This already yields the desired result, as we have: {displaystyle {begin{aligned}{text{A}}_{ABDE}+{text{A}}_{ACFG}&={text{A}}_{ABUH}+{text{A}}_{ACVH}\&={text{A}}_{BLQR}+{text{A}}_{RCMQ}\&={text{A}}_{BCML}end{aligned}}} References Howard Eves: Pappus's Extension of the Pythagorean Theorem.The Mathematics Teacher, Vol. 51, No. 7 (November 1958), pp. 544–546 (JSTOR) Howard Eves: Great Moments in Mathematics (before 1650). Mathematical Association of America, 1983, ISBN 9780883853108, p. 37 (excerpt, p. 37, at Google Books) Eli Maor: The Pythagorean Theorem: A 4,000-year History. Princeton University Press, 2007, ISBN 9780691125268, pp. 58–59 (excerpt, p. 58, at Google Books) Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 77–78 (excerpt, p. 77, at Google Books) External links The Pappus Area Theorem Pappus theorem 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 Wikimedia Commons has media related to Pappus's area theorem. Categories: AreaEquationsEuclidean plane geometryTheorems about triangles

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