mathematics known fundamental relation geometry sides right triangle states square opposite right angle equal sum squares sides written equation relating lengths sides called equation formula represents length lengths triangle sides argued knowledge named ancient greek mathematician tradition credited recorded proof evidence understood formula little indicates application mathematical framework indian chinese discovered independently cases provided special cases given numerous possibly mathematical diverse including geometric dating thousands years ways including higher dimensional spaces spaces objects right objects dimensional attracted interest outside mathematics symbol mathematical intellectual power popular references literature plays musicals songs stamps cartoons proof known long prove event proof attributed simple called proof large squares shown figure contain identical difference large squares arranged differently white space large squares equal area area white space yields originated simple proof writings later greek philosopher mathematician described known forms pointed introduction denotes length denote lengths sides expressed equation formula length known calculated formula length known length calculated formula formula equation relates sides right triangle simple way lengths sides known length found right triangle greater sides sum law allows length triangle given lengths sides angle angle sides right angle law reduces equation known law contender distinction book contains proof similar proof based sides similar fact ratio corresponding sides similar regardless size let abc represent right triangle right angle located shown figure draw altitude point intersection divides length parts new triangle similar triangle abc right angle definition altitude share angle meaning angle marked figure similar reasoning triangle similar abc proof similarity requires triangle sum angles triangle right angles equivalent parallel similarity leads equality corresponding sides formula result angles second result written formula results formula expresses formula role proof history subject speculation underlying question use proof invented proof similar involved theory proportions topic discussed later elements theory proportions needed development time proof outline proof elements proceeds large square divided left right triangle constructed half area left triangle constructed half area square left shown proving square area left argument followed similar version right remaining square putting reform square area sum area squares details follow let right triangle right angle perpendicular opposite square line divides square having area squares legs formal proof require elementary sides equal sides angles included sides equal angle area triangle half area base having altitude area equal product adjacent sides area square equal product sides follows square related triangle triangle related turn making lower square proof follows let right angled triangle right angle cab sides squares drawn order construction squares requires immediately preceding depends parallel draw line parallel intersect respectively join form bag right angles similarly right angles angle equals angle sum right angle angle abc equal equal triangle triangle straight line parallel twice area triangle share base altitude line normal common base connecting parallel lines square twice area triangle area square similarly shown area square adding results square proof appears elements proposition book demonstrates area square sum areas squares distinct proof similarity proof discussed proof proof idea conveyed animation consists large square containing identical right shown arrangements leaves squares uncovered second leaves square uncovered area encompassed outer square changes area beginning end black square areas equal second proof given middle animation large square formed area identical right sides fitted small central square formed sides moving combining smaller square produces squares areas area initial large square image gives proof upper squares divided shown blue green pieces rearranged fit lower square conversely large square divided shown pieces fill way cutting figure pieces figure called shows area large square equals smaller ones proof gave proof pieces need moved instead square squares legs use shape includes similar shapes include legs instead proof shape includes right triangle consists dropping perpendicular right angle triangle splitting triangle parts parts shape original right triangle legs original triangle sum areas original triangle ratio area right triangle square similar relationship areas holds squares sides large triangle proved copies right triangle sides arranged inside square half similar area formula small square area area large square formula square area formula similar proof uses copies triangle arranged square shown lower results larger square area square area larger square formula giving formula related proof published future president representative instead square uses constructed square second diagonal inner square shown area calculated half area square formula inner square similarly proof proceeds factor formula removed result proof arrive studying changes produce change employing triangle abc right triangle shown upper time triangle lengths measured shown length length length seen lower increased small extending slightly increases form sides triangle chosen perpendicular right triangle approximately similar abc sides formula rewritten formula differential equation solved direct integration formula giving formula constant equation formula intuitive proof formal rigorous proper limits place true positive numbers exists triangle sides triangle right angle sides lengths alternative statement triangle sides angle measures appears elements book proposition triangle square sides equals sum squares remaining sides triangle angle contained remaining sides triangle right proven law follows let abc triangle lengths construct second triangle sides length containing right angle follows triangle length triangle sides lengths angles angle lengths original triangle right angle proof makes use proven assuming simple means determining triangle right acute follows let chosen longest sides triangle according triangle inequality following statements apply triangle right triangle acute triangle stated proposition acute right language angle opposite angle opposite angle opposite sign function consequences uses triples triple positive words triple represents lengths sides right triangle sides lengths evidence monuments shows triples known discovery writing triple commonly written known examples primitive triple greatest common following list primitive triples values lengths consequences line segments lengths ratio rational number constructed compass enables construction lengths triangle related sides square root operation figure right shows construct line segments lengths ratio square root positive triangle labeled chosen unit measurement right triangle establishes length terms unit related unit square root positive perfect square realization length unit detail lengths conflicted school concept numbers numbers school dealt proportions comparison common according legend drowned sea making known existence complex numbers complex number formula absolute value given formula quantities related equation formula note defined positive number zero negative positive distance zero origin complex plane find distance points required distance given formula related version equation formula distance coordinate systems distance formula coordinates derived points plane distance called distance given formula generally space distance points formula formula defined formula coordinates example polar coordinates dimensions general terms coordinates expressing distance complicated derived typical example straight line distance points converted coordinates found applications physics discovered relating coordinates coordinates example polar coordinates introduced formula points locations separated distance formula performing squares combining terms formula distance coordinates produces separation polar coordinates formula product sum formula law called result case locations right angles enclosed angle form corresponding regained formula valid right special case general law valid arbitrary identity right triangle sides determines angle formula follows formula step applies relation called fundamental identity similar sides regardless size depend angles consequently figure triangle unit size opposite size sin adjacent size units relation cross product relates cross product dot product similar way formula seen definitions cross product dot product formula unit normal relationship follows definitions identity define cross product following equation obtained formula considered condition cross product definition example seven dimensions similar figures sides extending areas squares sides similar figures known 5th century included elements similar figures geometry corresponding sides sides right triangle sum areas ones smaller sides equals area larger extension assumes sides original triangle corresponding sides figures common sides similar figures proof applied convex applies concave similar figures curved boundaries figure boundary original triangle basic idea area plane figure proportional square linear dimension particular proportional square length similar figures areas erected sides corresponding lengths formula formula prove similar figures work backwards construct proof example starting center triangle replicated triangle similar right constructed sides formed dividing central triangle altitude sum areas smaller reversing logic leads law special case general relating lengths sides triangle law formula angle sides degrees formula reduces usual arbitrary triangle selected angle general triangle sides triangle equal angles base selected angle suppose selected angle opposite labeled triangle forms triangle angle opposite second triangle formed angle opposite length shown figure ibn stated sides related formula angle approaches base triangle narrows lengths overlap right triangle original regained proof observes triangle abc angles triangle opposite order share angle contain angle angle triangle consequently abc similar reflection triangle lower panel taking ratio sides opposite adjacent formula likewise reflection triangle formula clearing adding relations formula required result remains valid angle formula lengths non overlapping general area applies right sides place squares squares special case course upper figure shows triangle area longest sum areas sides provided long constructed indicated dimensions labeled arrows determine sides replacement squares bears clear resemblance original considered alexandria lower figure shows elements proof focus left figure left green area left blue portion base height left green area left green upper figure base upper left triangle height normal triangle repeating argument right figure area sum green solid geometry terms solid geometry applied dimensions follows consider rectangular solid shown figure length diagonal found formula sides form right triangle horizontal diagonal vertical edge length diagonal found second application formula step formula result dimensional expression magnitude diagonal terms components mutually perpendicular sides formula step viewed higher dimensions result repeated application original succession right sequence planes substantial dimensions named right angle corner like corner cube square area face opposite right angle corner sum squares areas faces result dimensional let formula consider dimensional formula think formula including origin remaining faces legs square volume sum squares volumes legs statement illustrated dimensions figure base figure legs sides foreground depth base increases area legs increases base fixed suggests depth value creating right applies different wording given rectangular dimensional square opposing right equal sum squares remaining inner product spaces inner product spaces familiar dimensional dimensional spaces example function considered infinitely components inner product space functional analysis inner product space concept replaced concept inner product formula zero inner product dot product dot product called standard inner product inner product inner products possible concept length replaced concept norm defined formula inner product space states formula akin sides right triangle given sum form consequence properties inner product formula inner products cross terms zero inner product space non law formula says twice sum squares lengths sides sum squares lengths norm equality facto norm corresponding inner product identity extended sums inner product space application successive pairs described dimensions section solid geometry results equation formula sets dimensional objects dimensional space introduced applies wide range objects sets objects number dimensions specifically square measure dimensional set objects parallel dimensional flats dimensional space equal sum squares measures projections object dimensional coordinate mathematical terms formula formula measure dimensions length dimension area dimensions volume dimensions etc formula set non overlapping dimensional objects parallel dimensional flats dimensional space formula total measure sum set dimensional objects formula represents dimensional projection original set coordinate formula measure dimensional set projection dimensional coordinate formula object projections overlap coordinate measure object projection set calculated individually measures projections added provide total measure set projections given coordinate formula number dimensional coordinate dimensional space dimensional objects projected formula example set dimensional parallel objects dimensional space coordinate calculation scenario coordinate planes plane plane plane required capture necessary projections calculating area set set contained dimensional parallel line segments instead coordinate axes planes needed capture projections calculating length set applied sets containing single object formula applied simplest case single dimensional object line segment dimensional space animation illustrates case line segment shown blue projections axes shown green lengths projections squared added equal length original line segment squared produces familiar formula formula length original line segment length segment projected axis length segment projected axis animation bringing line segment coordinate projections forms traditional right triangle similarly dimensional object dimensional space formula stated formula area specified dimensional object area object projection coordinate plane area object projection coordinate plane area object projection coordinate plane animation showing blue square object dimensions space illustrates application object dimension orientation object changes proportions green coordinate plane projections adjust accordingly squares areas projections add value square area original object case sum squares projection areas add applied sets containing multiple objects applies equally sets multiple objects long plane parallel planes measures objects set added essentially treated single object multiple line segment animation illustrates applied set dimensional objects dimensions space case line segments exist parallel line segment lines dimensional coordinate projected dimensional projections appear coordinate axes coordinate planes lengths projected line segments given axis summed squared added total lengths squared axes result squared sum lengths original line segments sake simplicity projections single points zero length shown affect calculations applies flat objects shape regular irregular multi object animation illustrates use set different objects different planes case triangle circle plane flat cat parallel plane shown blue projections set shown green coordinate plane objects shown initially upright plane subsequently parallel regardless set orientation result remains coordinate plane areas object projections calculated individually avoid projection overlap added produce total projection area set plane projection set area squared coordinate plane sum projection set areas squared equal original set area squared applied number dimensions holds regardless number dimensions involved volume squared dimensional object set calculated squares volumes associated dimensional projections dimensional number dimensions valid set long uses number dimensions coordinate projections built symmetry coordinate system coordinates unit length flat space allows apply broadly non geometry derived geometry fact given hold non geometry shown fact equivalent parallel fifth words non geometry relation sides triangle necessarily non form example spherical geometry sides right triangle unit sphere length equal angles right angles cases non geometry considered spherical geometry plane geometry case case non right result replacing follows appropriate law remains true geometry geometry condition triangle right replaced condition angles sum sides related follows sum areas circles equals area circle diameter spherical geometry right triangle sphere radius example figure right angle sides relation sides takes form formula equation derived special case spherical law applies spherical formula expressing series function expansion remainder term big formula shown radius approaches infinity arguments tend zero spherical relation sides right triangle approaches form expansion spherical relation right triangle yields formula absorbed big remainder terms independent radius relationship simplified quantities ones collecting error terms formula relationship recovered limit radius approaches infinity remainder term tends zero formula small right eliminated avoid loss significance giving formula geometry space uniform right triangle legs relation sides takes form formula formula special form law applies formula angle opposite series shown triangle small approach zero relation right triangle approaches form small right eliminated avoid loss significance giving formula small uniform positive zero negative small right shown formula differential geometry level dimensional space describes distance separated points formula element distance components separating points space called space geometry expression useful general coordinates general spaces takes form formula called metric abuse language term applied set function position describes curved space simple example flat space expressed coordinates example polar coordinates formula history debate discovered times places date discovery uncertain date proof according historian mathematics evidence indicates known 20th 16th centuries thousand years born scan tablet depicting proof widely images sources book mention discovered subsequently stated developed years born history divided parts knowledge triples knowledge relationship sides right triangle knowledge relationships adjacent angles system der triples discovered written includes problem solution triple problem mention triangle tablet written reign great contains entries closely related triples india dates given variously 8th 5th century contains list triples discovered statement proof right triangle contains numerical proof general area van der believed certainly based earlier traditions states influenced ancient math conclusive evidence favor opposition possibility contents known earlier surviving texts dating roughly 1st century chinese text classic gives reasoning triangle china called han dynasty triples appear chapters mathematical art mention right believe arose china alternatively known named duke mathematician reasoning composed dates commonly given methods construct triples according commentary wrote according specific exists surviving greek literature centuries lived authors attributed way suggests widely known formula attributed personally safely assume belongs oldest period mathematics according gave method finding triples combined geometry elements oldest extant proof presented popular culture arisen popular culture variety ways brief lives records years old looked geometry happened accidentally gentleman library elements lay open seventh proposition book read proposition said impossible reads demonstration referred proof referred read sic convinced truth love geometry wrote poem verse sullivan comic opera pirates lot news cheerful facts square makes oblique reference film wizard makes specific reference receiving diploma wizard immediately exhibits knowledge reciting incorrect version sum square roots sides triangle equal square root remaining joy got brain released coin shape right triangle coin tail image equation accompanied mention millennium greece japan issued postage stamps depicting speculative fiction referred geometric proof displayed alien ship demonstrate aliens understanding mathematics
Pythagorean theorem