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| [[Image:Euler diagram of triangle types.svg|thumb|320px|Position of some special triangles in an [[Euler diagram]] of types of triangles, using the definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.]]
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| A '''special right triangle''' is a [[right triangle]] with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45–45–90. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of [[Natural number|whole numbers]], such as 3 : 4 : 5, or of other special numbers such as the [[golden ratio]]. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
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| ==Angle-based==
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| [[File:Special right triangles for trig.svg|right|thumb|Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering [[trigonometric functions]] of multiples of 30 and 45 degrees.]]
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| "Angle-based" special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles.
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| The side lengths are generally deduced from the basis of the [[unit circle]] or other [[geometry|geometric]] methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.
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| Special triangles are used to aid in calculating common trigonometric functions, as below:
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| {| class="wikitable"
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| ! Degrees !! Radians !! sin !! cos !! tan
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| |-
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| | 0 || 0 || <math>\tfrac{\sqrt{0}}{2}=0</math> || <math>\tfrac{\sqrt{4}}{2}=1</math> || <math>0</math>
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| |-
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| | 30 || <math>\tfrac{\pi}{6}</math> || <math>\tfrac{\sqrt{1}}{2}=\tfrac{1}{2}</math> || <math>\tfrac{\sqrt{3}}{2}</math> || <math>\tfrac{1}{\sqrt{3}}</math>
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| |-
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| | 45 || <math>\tfrac{\pi}{4}</math> || <math>\tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}</math> || <math>\tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}</math> || <math>1</math>
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| |-
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| | 60 || <math>\tfrac{\pi}{3}</math> || <math>\tfrac{\sqrt{3}}{2}</math> || <math>\tfrac{\sqrt{1}}{2}=\tfrac{1}{2}</math> || <math>\sqrt{3}</math>
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| |-
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| | 90 || <math>\tfrac{\pi}{2}</math> || <math>\tfrac{\sqrt{4}}{2}=1</math> || <math>\tfrac{\sqrt{0}}{2}=0</math> || <math>\infty</math>
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| |}
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| {{multiple image
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| | width = 120
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| | image1 = Tile V488 bicolor.svg
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| | caption1 = 45–45–90
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| | image2 = Tile V46b.svg
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| | caption2 = 30–60–90
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| }}
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| The 45–45–90 triangle, the 30–60–90 triangle, and the equilateral/equiangular (60–60–60) triangle are the three [[Möbius triangle]]s in the plane, meaning that they [[tessellate]] the plane via reflections in their sides; see [[Triangle group]].
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| <br clear="all">
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| ===45–45–90 triangle===
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| [[Image:45-45-triangle.svg|thumb|150px|The side lengths of a 45–45–90 triangle]]
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| In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the [[Pythagorean theorem]].
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| Triangles with these angles are the only possible right triangles that are also [[isosceles triangle]]s in Euclidean geometry. However, in [[spherical geometry]] and [[hyperbolic geometry]], there are infinitely many different shapes of right isosceles triangles.
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| ===30–60–90 triangle===
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| [[Image:30-60-90.svg|thumb|150px|The side lengths of a 30–60–90 triangle]]
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| This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30°, 60°, and 90°. The sides are in the ratio 1 : √3 : 2.
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| The proof of this fact is clear using [[trigonometry]]. The [[geometry|geometric]] proof is:
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| :Draw an equilateral triangle ''ABC'' with side length 2 and with point ''D'' as the midpoint of segment ''BC''. Draw an altitude line from ''A'' to ''D''. Then ''ABD'' is a 30–60–90 triangle with hypotenuse of length 2, and base ''BD'' of length 1.
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| :The fact that the remaining leg ''AD'' has length √3 follows immediately from the [[Pythagorean theorem]].
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| The 30-60-90 triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α+δ, α+2δ are the angles in the progression then the sum of the angles 3α+3δ = 180°. So one angle must be 60° the other 90° leaving the remaining angle to be 30°.
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| ==Side-based==
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| {{Main|Pythagorean triple}}
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| Right triangles whose sides are of [[integer]] lengths, [[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> 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
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| :<math>m^2-n^2 : 2mn : m^2+n^2\,</math>
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| where ''m'' and ''n'' are any positive integers such that ''m''>''n''.
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| ===Common Pythagorean triples===
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| There are several Pythagorean triples which are well-known, including those with sides in the ratios:
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| :{| border="0" cellpadding="0" cellspacing="0"
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| !align="right"|3:||align="center"|4 ||align="left"|:5
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| |-
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| !align="right"|5:||align="center"|12||align="left"|:13
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| |-
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| !align="right"|8:||align="center"|15||align="left"|:17
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| |-
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| !align="right"|7:||align="center"|24||align="left"|:25
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| |-
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| !align="right"|9:||align="center"|40||align="left"|:41
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| |}
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| 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.
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| The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones, listed above) with both non-hypotenuse sides less than 256:
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| :{| border="0" cellpadding="0" cellspacing="0" align="left"
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| !align="right"|11:||align="center"|60||align="left"|:61||
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| |-
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| !align="right"|12:||align="center"|35||align="left"|:37
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| |-
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| !align="right"|13:||align="center"|84||align="left"|:85
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| |-
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| !align="right"|15:||align="center"|112||align="left"|:113
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| |-
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| !align="right"|16:||align="center"|63||align="left"|:65
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| |-
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| !align="right"|17:||align="center"|144||align="left"|:145
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| |-
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| !align="right"|19:||align="center"|180||align="left"|:181
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| |-
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| !align="right"|20:||align="center"|21||align="left"|:29
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| |-
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| !align="right"|20:||align="center"|99||align="left"|:101
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| |-
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| !align="right"|21:||align="center"|220||align="left"|:221
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| |}
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| {| border="0" cellpadding="0" cellspacing="0" align="left"
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| !align="right"|24:||align="center"|143||align="left"|:145||
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| |-
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| !align="right"|28:||align="center"|45||align="left"|:53
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| |-
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| !align="right"|28:||align="center"|195||align="left"|:197
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| |-
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| !align="right"|32:||align="center"|255||align="left"|:257
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| |-
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| !align="right"|33:||align="center"|56||align="left"|:65
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| |-
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| !align="right"|36:||align="center"|77||align="left"|:85
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| |-
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| !align="right"|39:||align="center"|80||align="left"|:89
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| |-
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| !align="right"|44:||align="center"|117||align="left"|:125
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| |-
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| !align="right"|48:||align="center"|55||align="left"|:73
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| |-
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| !align="right"|51:||align="center"|140||align="left"|:149
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| |}
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| {| border="0" cellpadding="0" cellspacing="0" align="left"
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| !align="right"|52:||align="center"|165||align="left"|:173||
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| |-
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| !align="right"|57:||align="center"|176||align="left"|:185
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| |-
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| !align="right"|60:||align="center"|91||align="left"|:109
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| |-
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| !align="right"|60:||align="center"|221||align="left"|:229
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| |-
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| !align="right"|65:||align="center"|72||align="left"|:97
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| |-
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| !align="right"|84:||align="center"|187||align="left"|:205
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| |-
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| !align="right"|85:||align="center"|132||align="left"|:157
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| |-
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| !align="right"|88:||align="center"|105||align="left"|:137
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| |-
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| !align="right"|95:||align="center"|168||align="left"|:193
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| |-
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| !align="right"|96:||align="center"|247||align="left"|:265
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| |}
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| {| border="0" cellpadding="0" cellspacing="0"
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| !align="right"|104:||align="center"|153||align="left"|:185
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| |-
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| !align="right"|105:||align="center"|208||align="left"|:233
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| |-
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| !align="right"|115:||align="center"|252||align="left"|:277
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| |-
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| !align="right"|119:||align="center"|120||align="left"|:169
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| |-
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| !align="right"|120:||align="center"|209||align="left"|:241
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| |-
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| !align="right"|133:||align="center"|156||align="left"|:205
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| |-
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| !align="right"|140:||align="center"|171||align="left"|:221
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| |-
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| !align="right"|160:||align="center"|231||align="left"|:281
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| |-
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| !align="right"|161:||align="center"|240||align="left"|:289
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| |-
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| !align="right"|204:||align="center"|253||align="left"|:325
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| |-
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| !align="right"|207:||align="center"|224||align="left"|:305
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| |}
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| <br clear="all">
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| ===Almost-isosceles Pythagorean triples===
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| Isosceles right-angled triangles cannot have sides with integer values. However, infinitely many ''almost-isosceles'' right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the [[Cathetus|non-hypotenuse edges]] differ by one.<ref>{{citation
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| | last1 = Forget | first1 = T. W.
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| | last2 = Larkin | first2 = T. A.
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| | issue = 3
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| | journal = Fibonacci Quarterly
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| | pages = 94–104
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| | title = Pythagorean triads of the form ''x'', ''x'' + 1, ''z'' described by recurrence sequences
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| | url = http://www.fq.math.ca/Scanned/6-3/6-3/forget.pdf
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| | volume = 6
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| | year = 1968}}.</ref><ref>{{citation
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| | last1 = Chen | first1 = C. C.
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| | last2 = Peng | first2 = T. A.
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| | journal = The Australasian Journal of Combinatorics
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| | mr = 1327342
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| | pages = 263–267
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| | title = Almost-isosceles right-angled triangles
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| | url = http://ajc.maths.uq.edu.au/pdf/11/ajc-v11-p263.pdf
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| | volume = 11
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| | year = 1995}}.</ref> Such almost-isosceles right-angled triangles can be obtained recursively,
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| :''a''<sub>0</sub> = 1, ''b''<sub>0</sub> = 2
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| :''a''<sub>''n''</sub> = 2''b''<sub>''n''–1</sub> + ''a''<sub>''n''–1</sub>''
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| :''b''<sub>''n''</sub> = 2''a''<sub>''n''</sub> + ''b''<sub>''n''–1</sub>''
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| ''a''<sub>''n''</sub> is length of hypotenuse, ''n'' = 1, 2, 3, .... Equivalently,
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| :<math>(\tfrac{x-1}{2})^2+(\tfrac{x+1}{2})^2 = y^2</math>
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| where {''x, y''} are the solutions to the [[Pell equation]] <math>x^2-2y^2 = -1</math>, 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>
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| :{| border="0" cellpadding="0" cellspacing="0"
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| !align="right"|3 : ||align="center"|4 ||align="left"| : 5
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| |-
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| !align="right"|20 : ||align="center"|21||align="left"| : 29
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| |-
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| !align="right"|119 : ||align="center"|120||align="left"| : 169
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| |-
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| !align="right"|696 : ||align="center"|697||align="left"| : 985
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| |-
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| !align="right"|4,059 : ||align="center"|4,060||align="left"| : 5,741
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| |-
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| !align="right"|23,660 : ||align="center"|23,661||align="left"| : 33,461
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| |-
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| !align="right"|137,903 : ||align="center"|137,904||align="left"| : 195,025
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| |-
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| !align="right"|803,760 : ||align="center"|803,761||align="left"| : 1,136,689
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| |-
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| !align="right"|4,684,659 : ||align="center"|4,684,660||align="left"| : 6,625,109
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| |}
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| Alternatively, the same triangles can be derived from the [[square triangular number]]s.<ref>{{citation
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| | last = Nyblom | first = M. A.
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| | issue = 4
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| | journal = The Fibonacci Quarterly
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| | mr = 1640364
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| | pages = 319–322
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| | title = A note on the set of almost-isosceles right-angled triangles
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| | url = http://www.fq.math.ca/Scanned/36-4/nyblom.pdf
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| | volume = 36
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| | year = 1998}}.</ref>
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| ===Right triangle whose sides are in a geometric progression===
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| [[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]]'''.]]
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| {{Main|Kepler triangle}}
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| The Kepler triangle is a right triangle whose sides are in a [[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'' = √''φ'' where ''φ'' is the golden ratio. Its sides are therefore in the ratio <math> 1:\sqrt{\varphi}:\varphi .\, </math>
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| ===Sides of regular polygons===
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| [[File:Euclid XIII.10.svg|thumb|The sides of a pentagon, hexagon, and decagon, inscribed in congruent circles, form a right triangle]]
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| Let <math>a=2\sin\tfrac{\pi}{10}=\tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\varphi}</math> be the side length of a regular [[decagon]] inscribed in a unit circle, where <math>\varphi</math> is the [[golden ratio]].
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| Let <math>b=2\sin\tfrac{\pi}{6}=1</math> be the side length of a regular [[hexagon]], and let <math>c=2\sin\tfrac{\pi}{5}=\sqrt{\tfrac{5-\sqrt{5}}{2}}</math> be the side length of a regular [[pentagon]]. Then <math>a^2+b^2=c^2</math>, so these three lengths form the sides of a right triangle.<ref>[http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html Euclid, 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>
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| ==See also==
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| * [[Integer triangle]]
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| * [[Spiral of Theodorus]]
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| * [[Triangle]]
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| ==References==
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| <references/>
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| ==External links==
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| * [http://www.mathopenref.com/triangle345.html 3 : 4 : 5 triangle]
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| * [http://www.mathopenref.com/triangle306090.html 30-60-90 triangle]
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| * [http://www.mathopenref.com/triangle454590.html 45-45-90 triangle] With interactive animations
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| {{DEFAULTSORT:Special Right Triangles}}
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| [[Category:Euclidean plane geometry]]
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| [[Category:Triangles]]
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