paaithaagoras prameya

is anuchhed ko vikipeediya lekh Pythagorean theorem ke is sanskaran se anuvaadit kiya gaya hai.

baudhaayan ka prameya: samakon tribhuj ki do bhujaaon ki lambaaiyon ke vargon ka yog karn ki lambaai ke varg ke baraabar hota hai.

paaithaagoras prameya (ya, baudhaayan prameya) yooklideeya jyaamiti mein kisi samakon tribhuj ke teenon bhujaaon ke beech ek sambandh bataane wala prameya hai. is prameya ko aamtaur par ek sameekaran ke roop mein nimnalikhit tareeke se abhivyakt kiya jaata hai-

${\displaystyle a^{2}+b^{2}=c^{2}\!\,}$

jahaaain c samakon tribhuj ke karn ki lanbaai hai tatha a aur b anya do bhujaaon ki lambaai hai. paaithaagoras yoonaan ke ganitjnya the. paramparaanusaar unhein hi is prameya ki khoj ka shreya diya jaata hai,^[1][1] haalaanki yeh maana jaane laga hai ki is prameya ki jaankaari unase poorv tithi ki hai. Bhaarat ke praacheen granth baudhaayan shulbasootr mein yeh prameya diya hua hai. kaafi pramaan hai ki bebeelon ke ganitjnyaon bhi is siddhaant ko jaante the. ise 'baudhaayan-paaithaagoras prameya' bhi kehte hain.

agar ham karn ki lanbaai ko c aur anya do bhujaaon ki lanbaai ko a aur b lete hain, to prameya ko nimnalikhit sameekaran ke roop mein vyakt kiya ja sakta hai:

${\displaystyle a^{2}+b^{2}=c^{2}\,}$

ya,

${\displaystyle c={\sqrt {a^{2}+b^{2}}}.\,}$

yadi c pehle se diya gaya hai aur ek bhuja ki lanbaai nikaalna ho, to nimnalikhit sameekaran ka upayog kiya ja sakta hai :

${\displaystyle c^{2}-a^{2}=b^{2}\,}$

ya

${\displaystyle c^{2}-b^{2}=a^{2}.\,}$

yeh sameekaran samakon trikon ke teenon bhujaaon ke beech ek saral sambandh pradaan karta hai. is prameya ka saamaanyeekaran 'kojya niyam' (Cosine rule) kahalaata hai jiski sahaayata se kisi bhi trikon ke teesari bhuja ki lambaai ki ganana ki ja sakti hai yadi shesh do bhujaaon ki lanbaai aur unke beech ke kon ki maap di gayi ho.

yeh ek aisa prameya hai jiske anya prameyon ki tulana mein sambhavat: sarvaadhik pramaan gyaat hain (dvighaati paarasparikta ka niyam bhi is gaurav ke liye pratiyogi rah chuka hai). eleesha Scott loomis dvaara rachit paayathaagauriyn thiaram kitaab mein, 367 pramaan diye gaye hain.

samaroop tribhuj ke upayog se pramaan

samaroop tribhuj ke upayog dvaara pramaan

baudhaayan prameya ke adhikaansh pramaanon ki tarah, yeh do samaroop tribhujon ki bhujaaon ke samaanupaati hone ke gun par aadhaarit hai.

maana ABC ek samakon tribhuj hai, jismein kon C samakon hai, jaisa aakruti mein dikhaaya gaya hai. ham C bindu se karn par lamb daalte hain aur bhuja AB ke saath us lamb ki lambaai H hain. yeh naya trikon ACH hamaare trikon ABC ke samaroop hai, kyonki un donon mein hi samakon hai (oonchaai ki paribhaasha ke dvaara) aur A kon unka hissa hai. iska matlab hai ki teesara kon bhi donon tribhujon mein samaan hai. isi aadhaar par tribhuj CBH bhi ABC ke samaroop hai. in samaroopataaon se hamein do samaanupaat praapt hote hain:

jaise

${\displaystyle BC=a,AC=b,{\text{ and }}AB=c,\!}$

tatha

${\displaystyle {\frac {a}{c}}={\frac {HB}{a}}{\mbox{ and }}{\frac {b}{c}}={\frac {AH}{b}}.\,}$

inhein aise bhi likha ja sakta hai

${\displaystyle a^{2}=c\times HB{\mbox{ and }}b^{2}=c\times AH.\,}$

in do sameekaranon ka sankshep karne par,

${\displaystyle a^{2}+b^{2}=c\times HB+c\times AH=c\times (HB+AH)=c^{2}.\,\!}$

anya shabdon mein, baudhaayan prameya:

${\displaystyle a^{2}+b^{2}=c^{2}.\,\!}$

yooklid ke pramaan

yooklid ke tatvon mein pramaan

yooklid ke tatvon mein, pustak 1 ka prastaav 47, baudhaayan prameya nimnalikhit laainon ke saath ek tark se saabit hota hai.A, B, C ko samakon trikon ke kone maanate hain, jismein samakon A par hoga. A se karn ke vipreet ek adholanb chhodein varg mein karn par.vo rekha karn par varg ko do aayaaton mein vibhaajit karti hai, pratyek ka samaan kshetr hai kyoonki donon mein se ek pairon mein varg banta hai.

aupachaarik pramaan ke liye, hamein chaar praathamik lemmata ki aavashyakta hai:

1. yadi do trikon ke do paarshvon mein se ek paarshv doosare ke do paarshvon ke baraabar ho, pratyek ke liye pratyek aur un paarshvon dvaara bana kon baraabar ho, to trikon anukool hain. (paarshv - kon - paarshv prameya)
2. ek trikon ka kshetrafal ek hi tal aur oonchaai par kisi bhi samaanaantar chaturbhuj ka aadha kshetrafal hai.
3. kisi bhi varg ka kshetrafal uske do paarshvon ke utpaad ke baraabar hota hai.
4. kisi bhi aayat ka kshetrafal uske do sanlagn paarshvon ke utpaad ke baraabar hota hai (lemma 3 se paalan karti hai).

is pramaan ke peechhe sahaj vichaar, jo iska paalan karna aasaan bana sakta hai, ki oopar ke do vargon ko ek hi aakaar ke samaanaantar chaturbhuj mein badla gaya hai, fir modkar aur baaen aur daahine aayat ko nichle varg mein badla gaya hai, fir nirantar kshetr mein.

nai laainein ko shaamil karke chitran

pramaan nimnaanusaar hai:

1. ACB ko samakon trikon maanate hain jismein samakon CAB hai.
2. pratyek paarshvon BC, AB aur CA mein, chauras banaaya gaya hai, CBDE, BAGF, and ACIH, is kram mein.
3. A se, BD aur CE karne ke liye ek samaanaantar rekha banaaeain. yeh lanbaroop mein BC aur DE ko K aur L mein kramash:, kaatata hai.
4. CF aur AD ko jodein, BCF aur BDA trikon banaane ke liye.
5. kon CAB aur BAG donon samakon hain; isliye C, A aur G ekarekhasth hain. isi prakaar bi, ke liye ek aur H.
6. kon CBD aur FBA donon samakon hain; isliye kon ABD kon FBC ke baraabar hai, kyoonki donon ek samakon aur kon ABC ke jod ke baraabar hain.
7. kyonki AB aur BD, FB and bak ke baraabar hain, kramash:, ABD trikon FBC trikon ke baraabar hona chaahiye.
8. kyoonki A, K aur L ke saath ekarekhasth hai, aayat BDLK ka kshetrafal ABD trikon se dugana hona chaahiye.
9. kyoonki C, A aur G ke saath ekarekhasth hai, varg BAGF ka kshetrafal FBC trikon se dugana hona chaahiye.
10. isliye aayat BDLK ka kshetrafal varg BAGF ke baraabar hona chaahiye = AB².
11. isi prakaar, yeh dikhaaya ja sakta hai ki aayat chakale ka kshetrafal varg ACIH ke baraabar hona chaahiye= AC².
12. in do parinaamon ko jodkar, AB² + AC² = BD × BK + KL × KC
13. kyoonki BD = KL, BD* BK + KL × KC = BD(BK + KC) = BD × BC
14. isliye AB AB² + AC² = BC², kyoonki CBDE ek varg hai.

yeh pramaan yooklid ke tatvon mein prastaav 1.47 ke roop mein pesh hota hai.^[2]

gaarafeeld ke pramaan

James A. gaarafeeld (paravarti sanyukt raajya America ke raashtrapati) ko ek upanyaas beejeeya pramaan dvaara shreya diya gaya hai:^[3]

poora samalamb (a+b) baai (a+b) varg ka aadha hai, to usaka kshetrafal = (a+b)²/2 = a²/2 + b²/2 + ab.

trikon 1 aur trikon 2 pratyek kshetrafal ab/2 hai.

trikon 3 ka kshetrafal c²/2 hai aur yeh karn par varg ka aadha hai.

lekin trikon 3 ka kshetrafal bhi = (samalamb ka kshetrafal) - (trikon 1 aur 2 ka kshetrafal)

= a²/2 + b²/2 + ab - ab/2 - ab/2

= a²/2 + b²/2

= anya do paarshvon ke vargon ke jod ka aadha hai.

isliye karn par varg = anya do paarshvon ke vargon ka jod hai.

vyavakalan dvaara pramaan

is pramaan mein, karn par varg plas trikon ki 4 pratiyaan ko anya do paarshvon mein vargon ke roop mein jod sakte hain plas trikon ki 4 pratiyaan.yeh pramaan cheen se darj ki gayi hai.

kshetr ghataav ke upayog dvaara pramaan

samaanata pramaan

oopar yooklid ke pramaan ke chitr se, ham teen samaan aankadon ko dekh sakte hain, pratyek mein "ek varg ke oopar trikon" hai. kyoonki bada trikon do chhote trikon se bana hai, usaka kshetrafal in do chhote ka jod hai. samaanata se, teen varg ek doosare ke saath usi anupaat mein hain jaise vah teen trikon aur isi tarah ke bade varg ka kshetrafal do chhote vargon ke kshetrafal ka jod hai.

vipryaya se pramaan

4 samaan samakon trikon ke punarnirmaan ke dvaara paayathaagauriyn prameya ke 101 pxpramaan: choonki kul kshetr aur trikon ke kshetr sabhi nirantar hain, kul kaala kshetr nirantar hai. lekin yeh varg vibhaajit kiya ja sakta hai a, b, c, paarshvon ke trikon se chitrit pradarshan ke dvaara [4] = c2.

vipryaya se pramaan ko chitran aur eneemeshan ke dvaara diya gaya hai. is udaaharan mein, har ek bade varg ka kshetrafal (a + b)² hai. donon mein, charon samaan trikon ka kshetrafal hata diya gaya hai. shesh kshetron, a² + b² aur c², baraabar hain.Q.E.D

enimeshan dvaara vipryaya se ek aur pramaan dikhaaya

vipryaya ka upayog karke pramaan

beejeeya pramaan: ek varg jo chaar samakon trikon aur ek bade varg ko shreneebaddh karke nirmit kiya hai

yeh pramaan vaastav mein bahut aasaan hai, lekin yeh praarambhik naheen hai, is arth mein ki yeh keval sabse buniyaadi siddhaant aur yukleediyn jyaamiti ke prameyon par nirbhar naheen hai. vishesh roop se, jab trikon aur vargon ke kshetrafal ka sootr dena bahut aasaan hai, yeh saabit karne ke liye aasaan naheen hai ki ek varg ka kshetrafal uske tukadon ke kshetron ka jod hai. vaastav mein, aavashyak gun saabit karna paayathaagauriyn prameya siddh karne ki tulana mein kathin hai (lebesgu upaaya aur banaach-taarski virodhaabhaas dekhein).vaastav mein, yeh kathinaai sabhi saadhaaran kshetr shaamil yukleediyn pramaan ko prabhaavit karta hai; udaaharan ke liye, ek samakon trikon ka kshetr paane ke liye ek dhaarana shaamil hai ki yeh ek hi oonchaai aur tal ke ek aayat ka aadha kshetr hai. isi kaaran se, jyaamiti ke liye svayansiddh parichay aam taur par trikon ki samaanata ke aadhaar par ek aur pramaan ka prayog karta hai (oopar dekhein).

is paayathaagauriyn prameya ka teesara graafik chitran mein (daahine mein peele aur neele rang mein) karn ka varg paarshvon ke varg mein fit baithata hai. ek sambandhit pramaan yeh dikha sakta hai ki pun: sthaapit bhaag mool ke samaan hain aur, kyoonki samaan ka jod samaan hai, ki unke kshetr bhi samaan hain. yeh dikhaane ke liye ki ek varg hi parinaam hai, hame nae paarshvon ki lanbaai ko c ke baraabar dikhaana padega.dhyaan dein ki is pramaan ke kaam karne ke liye, hame chhote varg ko aur adhik hisson mein kaatne ke tareeke ko sambhaalne ke liye raasta pradaan karna hoga choonki paarshv aur chhote hote jaaeainge.[4]

beejeeya pramaan

is pramaan ka beejeeya bhinnaroop nimn tark dvaara pradaan kiya gaya hai. chitran ko dekhte hue jo ek bada varg hai jiske konon mein samaan samakon trikon hai, in chaar trikon mein pratyek ka kshetr C ke saath ek kon ke dvaara diya gaya hai.

${\displaystyle {\frac {1}{2}}AB.}$

in trikon ke A-paarshv kon aur B-paarshv kon anupoorak kon hain, neele kshetr ke pratyek kon samakon hain, is kshetr ko ek varg banaate hue jiske paarshv ki lanbaai C hai. is varg ka kshetrafal hai C2.is tarah samast ka kshetr diya jaata hai:

${\displaystyle 4\left({\frac {1}{2}}AB\right)+C^{2}.}$

haalaanki, bade varg ke paarshvon ki lanbaai A + B[7], ham uske kshetrafal ki ganana kar sakte hain jaise (A + B)²[8], jo A² + 2AB + B²[9] mein vistaarit hota hai.

${\displaystyle A^{2}+2AB+B^{2}=4\left({\frac {1}{2}}AB\right)+C^{2}.\,\!}$

(4 ka vitran)${\displaystyle A^{2}+2AB+B^{2}=2AB+C^{2}\,\!}$

(2AB ka vyavakalan)${\displaystyle A^{2}+B^{2}=C^{2}\,\!}$

vibhedak sameekaranon dvaara pramaan

baudhaayan prameya mein pahuncha ja sakta hai nimnalikhit chitr ke adhyayan se ki ek paarshv mein parivartan kaise karn mein ek parivartan ke utpaadan kar sakta hai aur ek thoda kalan ka upayog karke.^[5]

vibhedak sameekaran ka upayog karke pramaan

paarshv a ke da mein parivartan ke parinaam svarup,

${\displaystyle {\frac {da}{dc}}={\frac {c}{a}}}$

trikon ki samaanata aur antar mein badlaav ke liye.isliye

${\displaystyle c\,dc=a\,da}$

char ke viyojan par.

paarshv b mein parivartan ke liye ek doosra kaaryakaal jodne ka parinaam hai.

samekit deta hai

${\displaystyle c^{2}=a^{2}+\mathrm {constant} .\ \,\!}$

jab a = 0 tab c = b, to b² "nirantar" hai. isliye

${\displaystyle c^{2}=a^{2}+b^{2}.\,}$

jaise dekha ja sakta hai, parivartan aur paarshvon ke beech vishesh anupaat ke kaaran hai yeh varg jabki paarshvon mein parivartan ki svatantr yogadaan ka parinaam raashi hai jo jyaamiteeya saakshyon se spasht naheen hai. is diye gaye anupaat se yeh dikhaaya ja sakta hai ki paarshvon mein parivartan paarshvon se prateepaanupaati anupaat hain. is vibhedak sameekaran sujhaav deta hai ki yeh prameya sambandhit parivartan ke kaaran hai aur iske vyutpatti lagbhag line abhinn abhiklan ke samaan hai.

yeh maatra da aur dc kramash: a aur c mein atyant chhote parivartan hain. lekin ham iske badle vaastavik sankhya Δaaa and Δaac ka upayog karte hain, tab unke anupaat ki seema da/dc hai jab unka aakaar shoonya niktata, vyutpanni aur c/a bhi niktata hai, trikon ke paarshvon ki lanbaai ka anupaat aur vibhedak sameekaran ka parinaam pata chalta hai.

is prameya ka vipryaaya bhi sach hai:

kisi bhi teen dhanaatmak sankhya a, b aur c aisi hai a² + b² = c²[10], vahaaain ek trikon maujood hai jiske paarshv hain a, b aur c aur har aise trikon mein paarshvon ke bheech ek samakon hai jinki lambaai a aur b hai.

yeh vipryaaya yooklid ke tatvon mein maujood hota hai.kosaain ki vidhi ka prayog karke yeh saabit kiya ja sakta hai (neeche dekhein saamaanyakaran ke neeche), ya nimnalikhit pramaan ke dvaara:

ABC ko ek trikon maanate hain jiske paarshvon ki lambaai a, b aur c hai, a² + b² = c² ke saath.hamein yeh saabit karna hai ki a aur b paarshvon ke beech ke kon samakon hai. ham ek aur trikon ka nirmaan karte hain jismein paarshvon ke beech ek samakon hai jiski lanbaai a aur b hai. paayathaagauriyn prameya se, nimnaanusaar hai ki is trikon ke karn lanbaai bhi c hai. choonki donon trikon ke paarshvon ki ek hi lanbaai hai a, b aur c, ve anukool hain aur isliye unka ek hi kon hona chaahiye.isliye, jin paarshvon ki lanbaai a aur b hai hamaare mool trikon mein unke beech ka kon ek samakon hai.

paayathaagauriyn prameya ke vipryaaya ka ek anumaan hai ki nirdhaaran karne ka ek saral tareeka hai ki yadi ek trikon samakon, obtyus, ya akyoot hai, is prakaar se.jahaaain c ko teenon paarshvon mein lamba chuna gaya hai:

• agar a² + b² = c², to trikon samakon hai.
• agar a² + b² > c², to trikon obtyus hai.
• agar a² + b² < c², to trikon akyoot hai.

paayathaagauriyn tripl

ek paayathaagauriyn tripl mein teen sakaaraatmak poornaank hain a, b aur c, jaise ki a² + b² = c².anya shabdon mein, ek paayathaagauriyn tripl ek samakon ke paarshvon ki lanbaai ka varnan karta hai jahaaain teenon paarshv ki poornaank lanbaai hai. uttari Europe ke bade pattharon se bane smaarakon se saakshya yeh dikhaate hain ki aise tripl likhne ki khoj se pehle se jaane jaante the. is tarah ke tripl saamaanyat: se likhe gaye hain (a, b, c).kuchh prasiddh udaaharan hain (3, 4, 5) aur (5, 12, 13)

aadim paayathaagauriyn ke tripl ki 100 tak ki soochi

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

tarkaheen sankhya ka astitv

paayathaagauriyn prameya ke parinaamon mein se ek hai ki taaratamyaheen lanbaai (ie. unke anupaat tarkaheen sankhya mein hai), jaise ki 2 ka vargamool, banaaya ja sakta hai. ek samakon jiske pair donon ek ikaai ke baraabar hain uske karn ki lanbaai 2 ka vargamool hai. yeh pramaan ki 2 ka vargamool tarkaheen hai lambe samay se aayojit vishvaas ke vipreet tha ki sab kuchh tarkasangat tha. pauraanik katha ke anusaar, hippaasus, jisne do ke vargamool ki tarkashoonyata sabse pehle saabit curry thi, use parinaam ke roop mein samudra mein doob gaya tha.^[6][7][8]

kaateejiyn nirdeshaank mein doorasth

kaateejiyn nirdeshaank mein doorasth formula ko paayathaagauriyn prameya se se praapt kiya gaya hai. agar (x₀, y₀) aur (x₁, y₁) chauras mein ank hain, to unke beech ki doori, jise yukleediyn doori bhi kaha jaata hai, jo diya jaata hai

${\displaystyle {\sqrt {(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}}.}$

atirikt saamaanyat: se, yukleediyn mein n-antar, do binduon ke beech ki yukleediyn doori, ${\displaystyle \scriptstyle A\,=\,(a_{1},a_{2},\dots ,a_{n})}$ aur ${\displaystyle \scriptstyle A\,=\,(a_{1},a_{2},\dots ,a_{n})}$, paayathaagauriyn prameya ka upayog karte hue, paribhaashit kiya gaya hai:

${\displaystyle {\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}$

samaan trikonon ke saamaanyakaran, hara [20] kshetr

yooklid ke tatvon mein baudhaayan prameya ko saamaanyakrut kiya gaya tha:

agar koi ek samaan aankade khada karta hai (yukleediyn jyaamiti dekhein) ek samakon trikon ke paarshvon mein, to do chhoton ke kshetron ka jod bade ke kshetrafal ke baraabar hai.

baudhaayan prameya, paarshvon ki lanbaai se sambandhit adhik saamaanya prameya ka ek vishesh case hai, kosaain ki vidhi:

${\displaystyle a^{2}+b^{2}-2ab\cos {\theta }=c^{2},\,}$

jahaan θ paarshvon a aur b ke beech ka kon hai.

jab θ 90 degree ho, to Cos(θ) = 0, to formula saamaanya baudhaayan prameya mein ban jaata hai.

is jatil aantarik utpaad antariksh mein do vektar v aur w diya jaae, to baudhaayan prameya nimnalikhit roop le leti hai:

${\displaystyle \|\mathbf {v} +\mathbf {w} \|^{2}=\|\mathbf {v} \|^{2}+\|\mathbf {w} \|^{2}+2\,{\mbox{Re}}\,\langle \mathbf {v} ,\mathbf {w} \rangle .}$

vishesh roop se,||v + w||² =||v||² +||w||² agar v aur w aayateeya hain, haalaanki vipryaaya ka sach hona zaroori naheen hai.

ganiteeya preran ka prayog karke, pichhla parinaam kisi parimit sankhya ke jodon mein aayateeya vektar tak badhaaya ja sakta hai. ke kisi bhi parimit sankhya ko badhaaya ja sakta hai.v₁, v₂,…, v_n ko vektar maanate hain ek aantarik utpaad antariksh mein jismein <v_i, vj> = 0 jab 1 ≤ i < j ≤ n.tab

${\displaystyle \left\|\,\sum _{k=1}^{n}\mathbf {v} _{k}\,\right\|^{2}=\sum _{k=1}^{n}\|\mathbf {v} _{k}\|^{2}.}$

is anant-aayaami ko asli aantarik utpaad sthaan ke parinaam ke saamaanyakaran ko paarseval ki pehchaan ke roop mein jaana jaata hai.

jab oopar ke prameya vektar ke baare mein thos jyaamiti mein pun: likha jaata hai, to yeh nimnalikhit prameya ban jaata hai. yadi AB aur BC rekhaaen B mein samakon banaate hain, BC aur kad rekhaaen C mein samakon banaate hain aur agar CD adholanb ke adholanb hai jismein AB aur BC rekhaaen shaamil hai, to AB, BC aur CD ki lambaai ke varg ka jod AD ke varg ke jod ke baraabar hai. yeh pramaan tuchh hai.

teen aayaamon ke liye baudhaayan prameya ka ek anya saamaanyakaran di guaa ka prameya hai, jo jeen paul di guaa de maalvs ke naam par rakha gaya hai: yadi ek tetraahedron mein ek samakon kon hai (ek ghan ki tarah ek kone), to varg ka kshetrafal jo samakon kone ke vipreet taraf hai vo anya teen taraf ke kshetron ke jode ke baraabar hai.

chaar aur adhik aayaamon mein in prameyon ke anuroop bhi hain .

jin trikon mein teen akyut kon hote hain, α + β > γ hota hai. isliye, a² + b² > c² hota hai.

jin trikon mein ek obtyus kon hota hai, α + β < γ hota hai. isliye, a² + b² < c² hota hai.

edsjar dijkstra ne is prastaav ko akyut, samakon aur obtyus trikon ke baare mein is bhaasha mein kaha hai:

sgn(α + β − γ) = sgn(a² + b² − c²)

jahaaain kon α paarshv a ke vipreet hai, kon β paarshv b ke vipreet hai aur kon γ paarshv c ke vipreet hai.^[9]

bina yukleediyn jyaamiti ke paayathaagauriyn prameya

yukleediyn jyaamiti ke siddhaant se paayathaagauriyn prameya se praapt hua hai, vaastav mein, oopar bataae paayathaagauriyn prameya ka yukleediyn prakaar bina yukleediyn jyaamiti ke naheen hota hai. (yeh vaastav mein yooklid ke samaantar (paanchavaan) svasiddh ke baraabar dikhaaya gaya hai.) udaaharan ke liye, goleeya jyaamiti mein, oktet se seemit ikaai kshetr ke samakon trikon ke teenon paarshvon ki lanbaai ${\displaystyle \scriptstyle \pi /2}$ ke baraabar hai; yukleediyn paayathaagauriyn prameya ka ullanghan karti hai kyoonki ${\displaystyle \scriptstyle \pi /2}$

iska matlab hai ki bina yukleediyn prameya mein, paayathaagauriyn prameya ko yukleediyn prameya se ek alag roop lena chaahiye.yahaaain do maamalon par vichaar karna padega- golaakaar jyaamiti aur atishyoktipoorn samatal jyaamiti hain; har maamale mein, yukleediyn maamale ki tarah, uchit kosaain ke niyam se parinaam niklata hai:

ek gola jiski trijya R hai usapar koi bhi samakon trikon ke liye, paayathaagauriyn prameya yeh roop leta hai

${\displaystyle \cos \left({\frac {c}{R}}\right)=\cos \left({\frac {a}{R}}\right)\,\cos \left({\frac {b}{R}}\right).}$

yeh sameekaran kosaain ke golaakaar kaanoon ka ek vishesh maamale ke roop mein praapt kiya ja sakta hai. is kosaain samaaroh ke liye maikalaurin shrrunkhala ka upayog karke, yeh dikhaaya ja sakta hai ki trijya R anantata tak pahunchata hai, ke roop mein hai, paayathaagauriyn prameya ka golaakaar roop yukleediyn roop tak pahunchata hai.

is atishyoktipoorn samatal mein kisi bhi trikon ke liye (gaussiyn vakrata -1 ke saath), paayathaagauriyn prameya yeh roop leta hai

${\displaystyle \cosh c=\cosh a\,\cosh b}$

jahaaain cosh ke atishyoktipoorn kosaain hai.

is prakaarya ke liye maikalaurin shrrunkhala ka upayog karke, yeh dikhaaya ja sakta hai ki jis tarah atishyoktipoorn trikon bahut chhoti ho jaat hai (arthaat jab a, b aur c shoonya niktate hain), paayathaagauriyn prameya ka atishyoktipoorn roop yukleediyn roop ko niktata hai.

atishyoktipoorn jyaamiti mein, ek samakon trikon ke liye bhi likha ja sakta hai,

${\displaystyle \sin {\bar {a}}\sin {\bar {b}}=\sin {\bar {c}}}$

jahaaain ${\displaystyle \scriptstyle {\bar {a}}}$ rekha khand ab ki samaanata ka kon jo ${\displaystyle \scriptstyle {\bar {a}}}$ jahaaain μ gunaatmak doori prakaarya hai (hilbart ke ant ke ankaganiteeya dekhein).

atishyoktipoorn trikonamiti mein, sign ke kon ki samaanata santusht karta hai

${\displaystyle \sin {\bar {a}}={\frac {2a}{1+a^{2}}}.}$

is prakaar, yeh sameekaran roop leta hai

${\displaystyle {\frac {2a}{1+a^{2}}}{\frac {2b}{1+b^{2}}}={\frac {2c}{1+c^{2}}}}$

jahaaain a, b, and c samakon trikon ke paarshvon ki ganaatmak dooriyaaain hain (haartashorn, 2000).

2 se adhik aayaamon mein

3 aayaamon mein ank { and { ke beech ki doori √([√((a-d)²+(b-e)²)]²+(c-f)²) = √((a-d)²+(b-d)²+(c-f)²) hai aur isi prakaar 4 ya adhik aayaamon ke liye.

jatil ankaganiteeya mein: maanya naheen

paayathaagauras formula ko kaarteejiyan nirdeshaank samatal mein do ankon ke beech ki doori pata karne ke liye prayog kiya jaata hai aur maanya hai agar sab nirdeshaank asli hain: ank {2+(b-d)²).lekin jatil nirdeshaank ke saath: udaaharan, ank { aur {i,0} ke beech ki doori shoonya banegi, jiska parinaam hai ridaakshiyo ed absurdam.yeh isliye hai kyonki yeh formula paayathaagauras ki prameya par nirbhar hai, jo apne has pramaan mein kshetrafal par nirbhar hai aur kshetrafal trikon par nirbhar hai aur anya jyaamiteeya aankadon par jo andar ko bahaar se alag karti hai, jo mumkin naheen hota agar nirdeshaank jatil hote.

is section mein satyaapan hetu atirikt sandarbh athva sroton ki aavashyakta hai. krupaya vishvasaneeya srot jodkar is section mein sudhaar karein. srotaheen saamagri ko chunauteedi ja sakti hai aur hataaya bhi ja sakta hai. (April 2008)

chau pi suaan ching 500-200 BC mein ke roop mein (3, 4, 5) trikon ka drushya pramaan

is prameya ka itihaas chaar bhaagon mein baaainta ja sakta hai: paayathaagauriyn tripl ka gyaan, samakon trikon paarshvon ke beech ke rishte ka gyaan, aasann kon ke beech sambandhon ke gyaan aur prameya ke pramaan.

misr mein bade pattharon ka bana smaarak lagbhag 2500 BC se aur uttari Europe mein, poornaank paarshvon ke samakon trikon shaamil hain.^[10] baartel leenadart vaun dr vaarden ka anumaan hai ki yeh paayathaagauriyn tripl ki khoj beejeeya se hui hai.^[11]

2000 aur 1786 BC ke beech likha gaya, misr ki madhyam kingadam paapirus barlin 6619 mein ek samasya shaamil hai jiska samaadhaan ek paayathaagauriyn tripl hai.

mesopotaamiya ke notabuk plimptan 322, 1790 aur 1750 BC mein mahaan haammurabi ke shaasanakaal ke dauraan likha gaya tha, jismein kai pravishtiyon shaamil hain jo paayathaagauriyn tripl ke niktata se sambandhit.

baudhayaanasulba sootr, jiski vibhinn taareek 8 veen shataabdi BC aur 2 veen shataabdi BC ke beech diye gaye hain, Bhaarat mein, jismein paayathaagauriyn tripl ki ek soochi shaamil hai jiski khoj beejeeya se hui hai, paayathaagauriyn prameya ka ek bayaan aur ek samadvibaahu samakon trikon ke liye paayathaagauriyn prameya ka jyaamitik pramaan hai.

apaastaamba sulba sootr (lagbhag 600 BC) mein saamaanya paayathaagauriyn prameya ki sankhyaatmak pramaan shaamil hain, ek kshetr sanganana ke upayog se.vaun dr vaarden ka vishvaas hai "yeh nishchit roop se pehle ke paranparaaon par aadhaarit thi".albart burk ke anusaar, yeh prameya ka mool pramaan hai; usane aage prameya kiya ki paayathaagauras ne aaraakonam ka daura kiya, Bhaarat aur usaki nakal curry.

paayathaagauras ne, jiski taareekhein saamaanyat: 569-475 BC di gayi hai, paayathaagauriyn tripl ke nirmaan ke liye beejeeya tareeke ka istemaal karke, yooklid mein proklos ki kameintri ke anusaar.proklos ne, tathaapi, 410 aur 485 AD ke beech likha tha. sar Thomas L. heeth ke anusaar, paayathaagauras ko prameya ka koi ropan naheen tha paaainch sadiyon tak paayathaagauras ke jeevit rahane tak.haalaanki, jab plootaarch aur siserau jaise lekhakon ne paayathaagauras ko prameya thaharaaya, unhonne is tarah se kiya jo ki ropan vyaapak roop se jaana jaae aur nissandeh rahe.^[1]

400 BC ke dauraan, proklos ke anusaar, pleto ne paayathaagauriyn tripl ko khojane ki ek vidhi di jise beejaganit aur jyaamiti sanghatit hua. lagbhag 300 BC, yooklid ke "tatvon" mein, prameya ka sabse puraana vartamaan siddhaanton wala pramaan pesh kiya gaya tha.

kuchh samay 500 BC aur 200 AD ke beech likha gaya tha, cheeni paath chau pi suan ching (周aa髀aa算aa经), (gnomon ke ankaganiteeya shaastreeya aur svarg ka pariptr raasta) paayathaagauriyn prameya ka ek drushya pramaan deta hai - cheen mein ise "gaugu prameya" kaha jaata hai (勾aa股aa定aa理) — trikon (3, 4, 5) ke liye.haan raajavansh ke dauraan, 202 BC se 220 AD tak, paayathaagauriyn tripl ko ganiteeya kala ke nauvein adhyaaya mein dekha gaya hai, samakon trikon ke ek ullekh ke saath.^[12]

cheen mein pehla record kiya gaya upayog hai, jo "gaugu prameya" (勾aa股aa定aa理) ke naam se jaana jaata hai, Bhaarat mein bhaaskar prameya ke naam se jaana jaata hai.

kaafi bahas hai ki kya paayathaagauriyn prameya ki khoj ek ya kai baar hui thi. boyar (1991) ka sochana hai ki shulba sootr mein paae gaye tatv mesopotaamiya vyutpatti ke ho sakte hain.^[13]

paayathaagauriyn prameya poore itihaas mein kai kism ki maas media mein sandarbhit hai.

• major-general ke sangeet ka ek padya gilbart aur suliven sangeetik penajains ke samudri daakoo, "dvipd prameya ke baare mein main bahut se samaachaar se bhara hua hooain, karn ke varg ke kai hansamukh tathyon ke saath", prameya ke tirchha sandarbh ke dvaara.
• [[vijrd of oj (1939 film)|vijrd of oj]] ka bijookha is prameya ka ek aur adhik vishisht sandarbh banaata hai jab use jaadoogar se diploma praapt hota hai. usane turant apne "gyaan" pradarshan ek vadh aur galat sanskaran padhne ke dvaara: "ek samadvibaahu trikon ke kisi bhi do paarshvon ke varg jadon ka jod shesh paarshvon ke varg jadon ke baraabar hai. oh, aanand, oh, umang.mujhemein dimaag hai

! "bijookha dvaara pradarshit "gyaan" galat hai. sahi bayaan "ek samakon trikon ke pairon ke vargon ka jod baaki paarshvon ke varg ke baraabar hain" hota.^[14]

• the sinpasans ke ek prakaran mein, henari kisinjar ke chashme ko springafeeld mein parmaanu oorja sanyantr ke shauchaalaya mein dhooaindhane ke baad, homar unhein pahanata hai aur uddharan karta hai oj bijookha ke sootr ka vadh sanskaran.paas mein ek shauchaalaya dukaan mein ek aadmi rokata hai aur chilaata hai "yeh ek samakon trikon hai, bevakoof

! "(varg jadon ke baare mein tippani kabhi sahi naheen hua.)

• isi tarah, Apple MacBook ki bhaashan software bijookha ke galat bayaan ko sandarbhit karta hai. yeh bhaashan ka namoona hai jab aavaaj seting raalf ko chuna jaata hai.
• sangataraashon mein, vigat ke master ka ek prateek yooklid ke 47 prastaav se ek chitr hai, paayathaagauriyn prameya ke yooklid ke pramaan mein prayukt.raashtrapati gaarafeeld ek sangataraash the.
• 2000 mein, yugaanda ne ek samakon trikon ke aakaar ka ek sikka jaari kiya. sikke ki pooainchh mein paithaagoras aur paayathaagauriyn prameya ka chitran tha, "paayathaagauras mileniym" ke ullekh ke saath.^[15] grees, Japan, sain mairino, siyra leon aur sooreenaam daak ticket jaari kiye hain paayathaagauras aur paayathaagauriyn prameya ke chitran ke saath.^[16]
• neel steefenasan ke vichaaravaan kalpana ainatham mein, paayathaagauriyn prameya ko "adraakhonik prameya" ke roop mein sandarbhit kiya gaya hai. is prameya ka ek jyaamitik pramaan ek videshi jahaaj ki ek taraf ganit ki unki samajh pradarshit karne ke liye pradarshit kiya.

vikimeediya kaumans par Pythagorean theorem se sambandhit media hai.

• paayathaagauriyn prameya (kat-di-not se 70 se adhik pramaan)
• intaraiktiv link:
  • jaava mein paayathaagauriyn prameya ka intaraektiv pramaan
  • [1]jaava mein paayathaagauriyn prameya ka ek aur intaraektiv pramaan
  • intaraiktiv eneemeshan ke saath paayathaagauriyn prameya
  • animated, ani-beejeeya aur upayogakarta-gati mein paayathaagauriyn prameya
• erik dablyoo veisateen, maithavarld par Pythagorean theorem
• paayathaagauriyn prameya aur samakon trikon ke faarmule