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[[অপারেটর (পদার্থবিজ্ঞান)]]
[[ব্রহ্মগুপ্ত ম্যাট্রিক্স]]
==Side-based==
Right triangles whose sides are of [[integer]] lengths, with the sides collectively known as [[Pythagorean triple]]s, possess angles that cannot all be [[rational numbers]] of [[Degree (angle)|degrees]].<ref>{{Cite journal|title=Rational Triangle|last=Weisstein|first=Eric W|journal=MathWorld|url=http://mathworld.wolfram.com/RationalTriangle.html}}</ref> (This follows from [[Niven's theorem]].) They are most useful in that they may be easily remembered and any [[multiple (mathematics)|multiple]] of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio
:{{nowrap|''m''{{sup|2}} − ''n''{{sup|2}} : 2''mn'' : ''m''{{sup|2}} + ''n''{{sup|2}}}}
where ''m'' and ''n'' are any positive integers such that {{nowrap|''m'' > ''n''}}.
===Common Pythagorean triples===
{{Main|Pythagorean triple}}
There are several Pythagorean triples which are well-known, including those with sides in the ratios:
:{| border="0" cellpadding="0" cellspacing="0"
!align="right"|3:||align="center"|4 ||align="left"|:5
|-
!align="right"|5:||align="center"|12||align="left"|:13
|-
!align="right"|8:||align="center"|15||align="left"|:17
|-
!align="right"|7:||align="center"|24||align="left"|:25
|-
!align="right"|9:||align="center"|40||align="left"|:41
|}
The 3 : 4 : 5 triangles are the only right triangles with edges in [[arithmetic progression]]. Triangles based on Pythagorean triples are [[Heronian triangle|Heronian]], meaning they have integer [[area]] as well as integer sides.
The possible use of the 3 : 4 : 5 triangle in [[Ancient Egypt]], with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated.<ref name=Cooke2011>{{cite book |last=Cooke |first=Roger L. |title=The History of Mathematics: A Brief Course |url=https://books.google.com/books?id=wOGh7XPowAMC |edition=2nd |year=2011|publisher=John Wiley & Sons |isbn=978-1-118-03024-0 |pages=237–238}}</ref> It was first conjectured by the historian [[Moritz Cantor]] in 1882.<ref name=Cooke2011/> It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement;<ref name=Cooke2011/> that [[Plutarch]] recorded in ''[[De Iside et Osiride|Isis and Osiris]]'' (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;<ref name=Cooke2011/> and that the [[Berlin Papyrus 6619]] from the [[Middle Kingdom of Egypt]] (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."<ref>{{cite book |author=Gillings, Richard J. |title=Mathematics in the Time of the Pharaohs |url=https://archive.org/details/mathematicsintim0000gill |url-access=registration |publisher=Dover |date=1982 |page=[https://archive.org/details/mathematicsintim0000gill/page/161 161]}}</ref> The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem."<ref name=Cooke2011/> Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".<ref name=Cooke2011/>
The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256:
:{| border="0" cellpadding="0" cellspacing="0" align="left"
!align="right"|11:||align="center"|60||align="left"|:61||
|-
!align="right"|12:||align="center"|35||align="left"|:37
|-
!align="right"|13:||align="center"|84||align="left"|:85
|-
!align="right"|15:||align="center"|112||align="left"|:113
|-
!align="right"|16:||align="center"|63||align="left"|:65
|-
!align="right"|17:||align="center"|144||align="left"|:145
|-
!align="right"|19:||align="center"|180||align="left"|:181
|-
!align="right"|20:||align="center"|21||align="left"|:29
|-
!align="right"|20:||align="center"|99||align="left"|:101
|-
!align="right"|21:||align="center"|220||align="left"|:221
|}
{| border="0" cellpadding="0" cellspacing="0" align="left"
!align="right"|24:||align="center"|143||align="left"|:145||
|-
!align="right"|28:||align="center"|45||align="left"|:53
|-
!align="right"|28:||align="center"|195||align="left"|:197
|-
!align="right"|32:||align="center"|255||align="left"|:257
|-
!align="right"|33:||align="center"|56||align="left"|:65
|-
!align="right"|36:||align="center"|77||align="left"|:85
|-
!align="right"|39:||align="center"|80||align="left"|:89
|-
!align="right"|44:||align="center"|117||align="left"|:125
|-
!align="right"|48:||align="center"|55||align="left"|:73
|-
!align="right"|51:||align="center"|140||align="left"|:149
|}
{| border="0" cellpadding="0" cellspacing="0" align="left"
!align="right"|52:||align="center"|165||align="left"|:173||
|-
!align="right"|57:||align="center"|176||align="left"|:185
|-
!align="right"|60:||align="center"|91||align="left"|:109
|-
!align="right"|60:||align="center"|221||align="left"|:229
|-
!align="right"|65:||align="center"|72||align="left"|:97
|-
!align="right"|84:||align="center"|187||align="left"|:205
|-
!align="right"|85:||align="center"|132||align="left"|:157
|-
!align="right"|88:||align="center"|105||align="left"|:137
|-
!align="right"|95:||align="center"|168||align="left"|:193
|-
!align="right"|96:||align="center"|247||align="left"|:265
|}
{| border="0" cellpadding="0" cellspacing="0"
!align="right"|104:||align="center"|153||align="left"|:185
|-
!align="right"|105:||align="center"|208||align="left"|:233
|-
!align="right"|115:||align="center"|252||align="left"|:277
|-
!align="right"|119:||align="center"|120||align="left"|:169
|-
!align="right"|120:||align="center"|209||align="left"|:241
|-
!align="right"|133:||align="center"|156||align="left"|:205
|-
!align="right"|140:||align="center"|171||align="left"|:221
|-
!align="right"|160:||align="center"|231||align="left"|:281
|-
!align="right"|161:||align="center"|240||align="left"|:289
|-
!align="right"|204:||align="center"|253||align="left"|:325
|-
!align="right"|207:||align="center"|224||align="left"|:305
|}
{{clear}}
===Almost-isosceles Pythagorean triples===
Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is {{sqrt|2}} and [[Square root of 2#Proofs of irrationality|{{sqrt|2}} cannot be expressed as a ratio of two integers]]. However, infinitely many ''almost-isosceles'' right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the [[Cathetus|non-hypotenuse edges]] differ by one.<ref>{{citation
| last1 = Forget | first1 = T. W.
| last2 = Larkin | first2 = T. A.
| issue = 3
| journal = Fibonacci Quarterly
| pages = 94–104
| title = Pythagorean triads of the form ''x'', ''x'' + 1, ''z'' described by recurrence sequences
| url = http://www.fq.math.ca/Scanned/6-3/6-3/forget.pdf
| volume = 6
| year = 1968}}.</ref><ref>{{citation
| last1 = Chen | first1 = C. C.
| last2 = Peng | first2 = T. A.
| journal = The Australasian Journal of Combinatorics
| mr = 1327342
| pages = 263–267
| title = Almost-isosceles right-angled triangles
| url = http://ajc.maths.uq.edu.au/pdf/11/ajc-v11-p263.pdf
| volume = 11
| year = 1995}}.</ref> Such almost-isosceles right-angled triangles can be obtained recursively,
:''a''<sub>0</sub> = 1, ''b''<sub>0</sub> = 2
:''a''<sub>''n''</sub> = 2''b''<sub>''n''−1</sub> + ''a''<sub>''n''−1</sub>''
:''b''<sub>''n''</sub> = 2''a''<sub>''n''</sub> + ''b''<sub>''n''−1</sub>''
''a''<sub>''n''</sub> is length of hypotenuse, ''n'' = 1, 2, 3, .... Equivalently,
:<math>(\tfrac{x-1}{2})^2+(\tfrac{x+1}{2})^2 = y^2</math>
where {''x'', ''y''} are solutions to the [[Pell equation]] {{nowrap|''x''{{sup|2}} − 2''y''{{sup|2}} {{=}} −1}}, with the hypotenuse ''y'' being the odd terms of the [[Pell numbers]] '''1''', 2, '''5''', 12, '''29''', 70, '''169''', 408, '''985''', 2378... {{OEIS|id=A000129}}.. The smallest Pythagorean triples resulting are:<ref>{{OEIS|A001652}}</ref>
:{| border="0" cellpadding="0" cellspacing="0"
!align="right"| 3 :|| align="center" | 4 ||align="left"|: 5
|-
!align="right"| 20 :|| align="center" | 21 ||align="left"|: 29
|-
!align="right"| 119 :|| align="center" | 120 ||align="left"|: 169
|-
!align="right"| 696 :|| align="center" | 697 ||align="left"|: 985
|-
!align="right"| 4,059 :|| align="center" | 4,060 ||align="left"|: 5,741
|-
!align="right"| 23,660 :|| align="center" | 23,661 ||align="left"|: 33,461
|-
!align="right"| 137,903 :|| align="center" | 137,904 ||align="left"|: 195,025
|-
!align="right"| 803,760 :|| align="center" | 803,761 ||align="left"|: 1,136,689
|-
!align="right"|4,684,659 :|| align="center" | 4,684,660 ||align="left"|: 6,625,109
|}
Alternatively, the same triangles can be derived from the [[square triangular number]]s.<ref>{{citation
| last = Nyblom | first = M. A.
| issue = 4
| journal = The Fibonacci Quarterly
| mr = 1640364
| pages = 319–322
| title = A note on the set of almost-isosceles right-angled triangles
| url = http://www.fq.math.ca/Scanned/36-4/nyblom.pdf
| volume = 36
| year = 1998}}.</ref>
===Arithmetic and geometric progressions===
[[File:Kepler triangle.svg|right|thumb|A '''Kepler triangle''' is a right triangle formed by three squares with areas in geometric progression according to the '''[[golden ratio]]'''.]]
{{Main article|Kepler triangle}}
The Kepler triangle is a right triangle whose sides are in [[geometric progression]]. If the sides are formed from the geometric progression ''a'', ''ar'', ''ar''<sup>2</sup> then its common ratio ''r'' is given by ''r'' = {{sqrt|''φ''}} where ''φ'' is the [[golden ratio]]. Its sides are therefore in the ratio {{nowrap|1 : {{sqrt|''φ''}} : ''φ''}}. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in geometric progression.
The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in [[arithmetic progression]].<ref>{{citation
| last1 = Beauregard | first1 = Raymond A.
| last2 = Suryanarayan | first2 = E. R.
| doi = 10.2307/2691431
| issue = 2
| journal = [[Mathematics Magazine]]
| mr = 1448883
| pages = 105–115
| title = Arithmetic triangles
| volume = 70
| year = 1997}}.</ref>
===Sides of regular polygons===
[[File:Euclid XIII.10.svg|thumb|The sides of a pentagon, hexagon, and decagon, inscribed in [[Congruence (geometry)|congruent]] circles, form a right triangle]]
Let {{nowrap|''a'' {{=}} 2 sin {{sfrac|{{pi}}|10}} {{=}} {{sfrac|−1 + {{sqrt|5}}|2}} {{=}} {{sfrac|1|''φ''}}}} be the side length of a regular [[decagon]] inscribed in the unit circle, where ''φ'' is the golden ratio. Let {{nowrap|''b'' {{=}} 2 sin {{sfrac|{{pi}}|6}} {{=}} 1}} be the side length of a regular [[hexagon]] in the unit circle, and let {{nowrap|''c'' {{=}} 2 sin {{sfrac|{{pi}}|5}} {{=}} <math>\sqrt{\tfrac{5-\sqrt{5}}{2}}</math>}} be the side length of a regular [[pentagon]] in the unit circle. Then {{nowrap|''a''{{sup|2}} + ''b''{{sup|2}} {{=}} ''c''{{sup|2}}}}, so these three lengths form the sides of a right triangle.<ref>[http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html Euclid's ''Elements'', Book XIII, Proposition 10].</ref> The same triangle forms half of a [[golden rectangle]]. It may also be found within a [[regular icosahedron]] of side length ''c'': the shortest line segment from any vertex ''V'' to the plane of its five neighbors has length ''a'', and the endpoints of this line segment together with any of the neighbors of ''V'' form the vertices of a right triangle with sides ''a'', ''b'', and ''c''.<ref>[http://ncatlab.org/nlab/show/pentagon+decagon+hexagon+identity nLab: pentagon decagon hexagon identity].</ref>
==See also==
| 