Editing Right triangle (section)

==Characterizations==
A triangle <math>\triangle ABC</math> with sides <math>a \le b < c</math>, [[semiperimeter]] <math display=inline>s = \tfrac12(a+b+c)</math>, [[area]] <math>T,</math> [[altitude (triangle)|altitude]] <math>h_c</math> opposite the longest side, [[Circumscribed circle|circumradius]] <math>R,</math> [[Incircle and excircles of a triangle#Relation to area of the triangle|inradius]] <math>r,</math> [[Incircle and excircles of a triangle#Relation to area of the triangle|exradii]] <math>r_a, r_b, r_c</math> tangent to <math>a,b,c</math> respectively, and [[median (geometry)|medians]] <math>m_a, m_b, m_c</math> is a right triangle [[if and only if]] any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle.

===Sides and semiperimeter===
* <math>a^2+b^2=c^2\quad (\text{Pythagorean theorem})</math>
* <math>(s-a)(s-b) = s(s-c)</math>
* <math>s=2R+r.</math><ref>{{Cite web |url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=411120 |title=Triangle right iff s = 2R + r, ''Art of problem solving'', 2011 |access-date=2012-01-02 |archive-date=2014-04-28 |archive-url=https://web.archive.org/web/20140428221212/http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=411120 |url-status=dead }}</ref>
* <math>a^2+b^2+c^2=8R^2.</math><ref name=Andreescu/>

===Angles===
* <math>A</math> and <math>B</math> are [[complementary angles|complementary]].<ref>{{Cite web |url=http://www.ricksmath.com/right-triangles.html |title=Properties of Right Triangles |access-date=2012-02-15 |archive-date=2011-12-31 |archive-url=https://web.archive.org/web/20111231222001/http://www.ricksmath.com/right-triangles.html |url-status=dead }}</ref>
* <math>\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] {{Webarchive|url=https://web.archive.org/web/20130805051705/http://www.cut-the-knot.org/wiki-math/index.php?n=Trigonometry.AVariantOfPythagoreanTheorem|date=2013-08-05}}.</ref>
* <math>\sin^2{A}+\sin^2{B}+\sin^2{C}=2.</math><ref name=Andreescu/><ref name=CTK/>
* <math>\cos^2{A}+\cos^2{B}+\cos^2{C}=1.</math><ref name=CTK/>
* <math>\sin{2A}=\sin{2B}=2\sin{A}\sin{B}.</math>

===Area===
* <math>T=\frac{ab}{2}</math>
* <math>T=r_ar_b=rr_c</math>
* <math>T=r(2R+r)</math>
* <math>T=\frac{(2s-c)^2-c^2}{4}=s(s-c)</math>
* <math>T=|PA| \cdot |PB|,</math> where <math>P</math> is the tangency point of the [[Incircle and excircles of a triangle|incircle]] at the longest side <math>AB.</math><ref>{{citation
|last=Darvasi |first=Gyula
|journal=The Mathematical Gazette
|pages=72–76
|title=Converse of a Property of Right Triangles
|volume=89
|number=514
|date=March 2005|doi=10.1017/S0025557200176806
|s2cid=125992270
|doi-access=free
}}.</ref>

===Inradius and exradii===
* <math>r=s-c=(a+b-c)/2</math>
* <math>r_a=s-b=(a-b+c)/2</math>
* <math>r_b=s-a=(-a+b+c)/2</math>
* <math>r_c=s=(a+b+c)/2</math>
* <math>r_a+r_b+r_c+r=a+b+c</math>
* <math>r_a^2+r_b^2+r_c^2+r^2=a^2+b^2+c^2</math>
* <math>r=\frac{r_ar_b}{r_c}.</math><ref name=Bell>{{citation|last=Bell |first=Amy|journal=Forum Geometricorum|pages=335–342|title=Hansen's Right Triangle Theorem, Its Converse and a Generalization|url=http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |archive-url=https://web.archive.org/web/20080725014729/http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |archive-date=2008-07-25 |url-status=live|volume=6|year=2006}}</ref>

===Altitude and medians===
{{right_angle_altitude.svg}}
* <math>h_c=\frac{ab}{c}</math>
* <math>m_a^2+m_b^2+m_c^2=6R^2.</math><ref name=Crux/>{{rp|Prob. 954, p. 26}}
* The length of one [[Median (geometry)|median]] is equal to the [[Circumscribed circle|circumradius]].
* 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]].

===Circumcircle and incircle===
* The triangle can be inscribed in a [[semicircle]], with one side coinciding with the entirety of the diameter ([[Thales' theorem]]).
* The [[Circumscribed circle|circumcenter]] is the [[midpoint]] of the longest side.
* The longest side is a [[diameter]] of the [[Circumscribed circle#Circumscribed circles of triangles|circumcircle]] <math>(c=2R).</math>
* 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>
* The [[Altitude (triangle)#Orthocenter|orthocenter]] lies on the circumcircle.<ref name=Crux/>
* 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/>