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| [[Image:Rtriangle.svg|thumb|right|Right triangle]]
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| A '''right triangle''' ([[American English]]) or '''right-angled triangle''' ([[British English]]) is a [[triangle]] in which one [[angle]] is a [[right angle]] (that is, a 90-[[Degree (angle)|degree]] angle). The relation between the sides and angles of a right triangle is the basis for [[trigonometry]].
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| The side opposite the right angle is called the ''[[hypotenuse]]'' (side ''c'' in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: ''[[wiktionary:cathetus|cathetus]]''). Side ''a'' may be identified as the side ''adjacent to angle B'' and ''opposed to'' (or ''opposite'') ''angle A'', while side ''b'' is the side ''adjacent to angle A'' and ''opposed to angle B''.
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| If the lengths of all three sides of a right triangle are integers, the triangle is said to be a '''Pythagorean triangle''' and its side lengths are collectively known as a [[Pythagorean triple]].
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| ==Principal properties==
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| ===Area===
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| As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area ''T'' is
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| :<math>T=\tfrac{1}{2}ab</math>
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| where ''a'' and ''b'' are the legs of the triangle.
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| If the [[Incircle and excircles of a triangle|incircle]] is tangent to the hypotenuse AB at point P, then denoting the semi-perimeter {{nowrap|(''a'' + ''b'' + ''c'') / 2}} as ''s'', we have {{nowrap|PA {{=}} ''s'' − ''a''}} and {{nowrap|PB {{=}} ''s'' − ''b''}}, and the area is given by
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| :<math>T=\text{PA} \cdot \text{PB} = (s-a)(s-b).</math>
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| This formula only applies to right triangles.<ref>Di Domenico, Angelo S., "A property of triangles involving area", ''[[Mathematical Gazette]]'' 87, July 2003, pp. 323-324.</ref>
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| ===Altitudes===
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| [[Image:Teorema.png|thumb|right|Altitude of a right triangle]]<!--Better diagram needed here.-->
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| If an [[Altitude (triangle)|altitude]] is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both [[Similarity (geometry)|similar]] to the original and therefore similar to each other. From this:
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| *The altitude to the hypotenuse is the [[geometric mean]] ([[Ratio#Euclid's definitions|mean proportional]]) of the two segments of the hypotenuse.
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| *Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
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| In equations,
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| :<math>\displaystyle f^2=de,</math> (this is sometimes known as the [[right triangle altitude theorem]])
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| :<math>\displaystyle b^2=ce,</math>
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| :<math>\displaystyle a^2=cd</math>
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| where ''a'', ''b'', ''c'', ''d'', ''e'', ''f'' are as shown in the diagram.<ref>Wentworth p. 156</ref> Thus
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| :<math>f=\frac{ab}{c}.</math>
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| Moreover, the altitude to the hypotenuse is related to the legs of the right triangle by<ref>Voles, Roger, "Integer solutions of <math>a^{-2} + b^{-2} = d^{-2}</math>," ''Mathematical Gazette'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–317.</ref>
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| :<math>\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{f^2}.</math>
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| The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's [[orthocenter]]—the intersection of its three altitudes—coincides with the right-angled vertex.
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| ===Pythagorean theorem===
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| {{main|Pythagorean theorem}}
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| The [[Pythagorean theorem]] states that:
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| <blockquote>In any right triangle, the area of the [[Square (geometry)|square]] whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).</blockquote>
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| This can be stated in equation form as
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| :<math>\displaystyle a^2+b^2=c^2</math>
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| where ''c'' is the length of the hypotenuse, and ''a'' and ''b'' are the lengths of the remaining two sides.
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| ===Inradius and circumradius===
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| [[File:Illustration to Euclid's proof of the Pythagorean theorem.png|thumb|200x200px|Illustration of the '''Pythogorean Theorem''']]
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| The radius of the [[incircle]] of a right triangle with legs ''a'' and ''b'' and hypotenuse ''c'' is
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| :<math>r = \frac{a+b-c}{2} = \frac{ab}{a+b+c}.</math>
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| The radius of the [[circumcircle]] is half the length of the hypotenuse,
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| :<math>R = \frac{c}{2}.</math>
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| One of the legs can be expressed in terms of the inradius and the other leg as
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| :<math>\displaystyle a=\frac{2r(b-r)}{b-2r}.</math>
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| ==Characterizations==
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| A triangle ''ABC'' with sides <math>a \le b < c</math>, [[semiperimeter]] ''s'', [[area]] ''T'', [[altitude]] ''h'' opposite the longest side, [[Circumscribed circle|circumradius]] ''R'', [[Incircle and excircles of a triangle#Relation to area of the triangle|inradius]] ''r'', [[Incircle and excircles of a triangle#Relation to area of the triangle|exradii]] ''r<sub>a</sub>'', ''r<sub>b</sub>'', ''r<sub>c</sub>'' (tangent to ''a'', ''b'', ''c'' respectively), and [[Median (geometry)|medians]] ''m<sub>a</sub>'', ''m<sub>b</sub>'', ''m<sub>c</sub>'' is a right triangle [[if and only if]] any one of the statements in the following six categories is true. All of them are of course also properties of a right triangle, since characterizations are equivalences.
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| ===Sides and semiperimeter===
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| *<math>\displaystyle a^2+b^2=c^2\quad (\text{Pythagoras})</math>
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| *<math>\displaystyle (s-a)(s-b)=s(s-c)</math>
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| *<math>\displaystyle s=2R+r.</math><ref>Triangle right iff s = 2R + r, ''Art of problem solving'', 2011, [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=411120]</ref>
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| *<math>\displaystyle a^2+b^2+c^2=8R^2.</math><ref name=Andreescu/>
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| ===Angles===
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| *''A'' and ''B'' are [[complementary angles|complementary]].<ref>[http://www.ricksmath.com/right-triangles.html Properties of Right Triangles]</ref>
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| *<math>\displaystyle \cos{A}\cos{B}\cos{C}=0.</math><ref name=Andreescu/><ref name=CTK>CTK Wiki Math, ''A Variant of the Pythagorean Theorem'', 2011, [http://www.cut-the-knot.org/wiki-math/index.php?n=Trigonometry.AVariantOfPythagoreanTheorem].</ref>
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| *<math>\displaystyle \sin^2{A}+\sin^2{B}+\sin^2{C}=2.</math><ref name=Andreescu/><ref name=CTK/>
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| *<math>\displaystyle \cos^2{A}+\cos^2{B}+\cos^2{C}=1.</math><ref name=CTK/>
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| *<math>\displaystyle \sin{2A}=\sin{2B}=2\sin{A}\sin{B}.</math>
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| ===Area===
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| *<math>\displaystyle T=\frac{ab}{2}</math>
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| *<math>\displaystyle T=r_ar_b=rr_c</math>
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| *<math>\displaystyle T=r(2R+r)</math>
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| *<math>T=PA\cdot PB,</math> where ''P'' is the tangency point of the [[Incircle and excircles of a triangle|incircle]] at the longest side ''AB''.<ref>{{citation
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| |last=Darvasi |first=Gyula
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| |journal=The Mathematical Gazette
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| |pages=72–76
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| |title=Converse of a Property of Right Triangles
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| |volume=89
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| |number=514
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| |date=March 2005}}.</ref>
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| ===Inradius and exradii<ref name=Bell>{{citation
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| |last=Bell |first=Amy
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| |journal=Forum Geometricorum
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| |pages=335–342
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| |title=Hansen's Right Triangle Theorem, Its Converse and a Generalization
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| |url=http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf
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| |volume=6
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| |year=2006}}.</ref>===
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| *<math>\displaystyle r=s-c</math>
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| *<math>\displaystyle r_a=s-b</math>
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| *<math>\displaystyle r_b=s-a</math>
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| *<math>\displaystyle r_c=s</math>
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| *<math>\displaystyle r_a+r_b+r_c+r=a+b+c</math>
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| *<math>\displaystyle r_a^2+r_b^2+r_c^2+r^2=a^2+b^2+c^2</math>
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| *<math>\displaystyle r=\frac{r_ar_b}{r_c}</math>
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| ===Altitude and medians===
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| *<math>\displaystyle h=\frac{ab}{c}</math>
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| *<math>\displaystyle m_a^2+m_b^2+m_c^2=6R^2.</math><ref name=Crux>''Inequalities proposed in “Crux Mathematicorum”'', Problem 954, p. 26, [http://www.imomath.com/othercomp/Journ/ineq.pdf].</ref>
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| *The length of one [[Median (geometry)|median]] is equal to the [[Circumscribed circle|circumradius]].
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| *The shortest [[Altitude (triangle)|altitude]] (the one from the vertex with the biggest angle) is the [[geometric mean]] of the [[line segment]]s it divides the opposite (longest) side into. This is the [[right triangle altitude theorem]].
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| ===Circumcircle and incircle===
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| *The triangle can be inscribed in a [[semicircle]], with one side coinciding with the entirety of the diameter ([[Thales' theorem]]).
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| *The [[Circumscribed circle|circumcenter]] is the [[midpoint]] of the longest side.
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| *The longest side is a [[diameter]] of the [[Circumscribed circle#Circumscribed circles of triangles|circumcircle]] <math>\displaystyle (c=2R).</math>
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| *The circumcircle is [[tangent]] to the [[nine-point circle]].<ref name=Andreescu>Andreescu, Titu and Andrica, Dorian, "Complex Numbers from A to...Z", Birkhäuser, 2006, pp. 109-110.</ref>
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| *The [[Altitude (triangle)#The orthocenter|orthocenter]] lies on the circumcircle.<ref name=Crux/>
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| *The distance between the [[Incircle and excircles of a triangle|incenter]] and the orthocenter is equal to <math>\sqrt{2}r</math>.<ref name=Crux/>
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| ==Trigonometric ratios==
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| {{main|Trigonometric functions#Right-angled triangle definitions|l1=Trigonometric functions – Right-angled triangle definitions}}
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| The [[trigonometric functions]] for acute angles can be defined as ratios of the sides of a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are [[similar triangles|similar]]. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled ''O'', ''A'' and ''H'' respectively, then the trigonometric functions are
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| :<math>\sin\alpha =\frac {O}{H},\,\cos\alpha =\frac {A}{H},\,\tan\alpha =\frac {O}{A},\,\sec\alpha =\frac {H}{A},\,\cot\alpha =\frac {A}{O},\,\csc\alpha =\frac {H}{O}.</math>
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| ==Special right triangles==
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| {{main|Special right triangles}}
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| The values of the trigonometric functions can be evaluated exactly for certain angles using right triangles with special angles. These include the ''30-60-90 triangle'' which can be used to evaluate the trigonometric functions for any multiple of π/6, and the ''45-45-90 triangle'' which can be used to evaluate the trigonometric functions for any multiple of π/4.
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| Right triangle first one whose angles are in a geometric progression and other one is sides are in arithmetic progression like (3,4,5).
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| The [[hyperbolic triangle#Euclidean geometry|hyperbolic triangle]] is a special right triangle used to define the [[hyperbolic function]]s.
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| ==Thales' theorem==
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| {{main|Thales' theorem}}
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| [[Image:thm mediane.svg|thumb|300px|right|Median of a right angle of a triangle]]
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| '''Thales' theorem''' states that if ''A'' is any point of the circle with diameter ''BC'' (except ''B'' or ''C'' themselves) ''ABC'' is a right triangle where ''A'' is the right angle. The converse states that if a right triangle is inscribed in a circle then the hypotenuse will be a diameter of the circle. A corollary is that the length of the hypotenuse is twice the distance from the right angle vertex to the midpoint of the hypotenuse. Also, the center of the circle that [[Circumscribed circle|circumscribes]] a right triangle is the midpoint of the hypotenuse and its radius is one half the length of the hypotenuse.
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| ==Medians==
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| The following formulas hold for the [[median (geometry)|median]]s of a right triangle:
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| :<math>m_a^2 + m_b^2 = 5m_c^2 = \frac{5}{4}c^2.</math>
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| The median on the hypotenuse of a right triangle divides the triangle into two isosceles triangles, because the median equals one-half the hypotenuse.
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| ==Relation to various means and the golden ratio==
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| Let ''H'', ''G'', and ''A'' be the [[harmonic mean]], the [[geometric mean]], and the [[arithmetic mean]] of two positive numbers ''a'' and ''b'' with ''a'' > ''b''. If a right triangle has legs ''H'' and ''G'' and hypotenuse ''A'', then<ref>Di Domenico, A., "The golden ratio — the right triangle — and the arithmetic, geometric, and harmonic means," ''Mathematical Gazette'' 89, July 2005, 261. Also Mitchell, Douglas W., "Feedback on 89.41", vol 90, March 2006, 153-154.</ref>
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| :<math>\frac{A}{H} = \frac{A^{2}}{G^{2}} = \frac{G^{2}}{H^{2}} = \phi \,</math>
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| and
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| :<math>\frac{a}{b} = \phi^{3}, \, </math>
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| where <math>\phi</math> is the [[golden ratio]] <math>\tfrac{1+ \sqrt{5}}{2}. \,</math>
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| ==Euler line==
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| In a right triangle, the [[Euler line]] contains the median on the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its [[Bisection#Perpendicular bisectors|perpendicular bisectors of sides]], falls on the midpoint of the hypotenuse.
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| ==Other properties==
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| If segments of lengths ''p'' and ''q'' emanating from vertex ''C'' trisect the hypotenuse into segments of length ''c''/3, then<ref>Posamentier, Alfred S., and Salkind, Charles T. ''Challenging Problems in Geometry'', Dover, 1996.</ref>{{rp|pp. 216–217}}
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| :<math>p^2 + q^2 = 5\left(\frac{c}{3}\right)^2.</math>
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| The right triangle is the only triangle having two, rather than three, distinct inscribed squares.<ref>Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles", ''[[Mathematics Magazine]]'' 71(4), 1998, 278-284.</ref>
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| Let ''h'' and ''k'' (''h'' > ''k'') be the sides of the two inscribed squares in a right triangle with hypotenuse ''c''. Then
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| :<math>\frac{1}{c^2} + \frac{1}{h^2} = \frac{1}{k^2}.</math>
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| These sides and the incircle radius ''r'' are related by a similar formula:
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| :<math>\displaystyle \frac{1}{r}=-{\frac{1}{c}}+\frac{1}{h}+\frac{1}{k}.</math>
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| The perimeter of a right triangle equals the sum of the radii of [[Incircle|the incircle and the three excircles]].
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| ==References==
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| {{reflist}}
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| *{{MathWorld |title=Right Triangle |urlname=RightTriangle}}
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| *{{cite book |title=A Text-Book of Geometry |first=G.A. |last=Wentworth |publisher=Ginn & Co. |year=1895 |url=http://www.archive.org/details/atextbookgeomet10wentgoog}}
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| ==External links==
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| {{Commons category|Right triangles}}
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| *[http://www.kurztutorial.info/mathematik/trigonometrie/en/dreieck.html Calculator for right triangles]
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| *[http://www.hackmath.net/en/calculator/triangle-calculator?what=rt Advanced right triangle calculator]
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| [[Category:Triangles]]
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| [[Category:Triangle geometry]]
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