Editing Right triangle (section)

==Principal properties==

===Sides===
[[File:Illustration to Euclid's proof of the Pythagorean theorem2.svg|thumb|The diagram for Euclid's proof of the Pythagorean theorem: each smaller square has area equal to the rectangle of corresponding color.]]
{{main|Pythagorean theorem}}

The three sides of a right triangle are related by the [[Pythagorean theorem]], which in modern algebraic notation can be written

:<math>a^2 + b^2 = c^2,</math>

where <math>c</math> is the length of the ''hypotenuse'' (side opposite the right angle), and <math>a</math> and <math>b</math> are the lengths of the ''legs'' (remaining two sides). [[Pythagorean triple]]s are integer values of <math>a, b, c</math> satisfying this equation. This theorem was proven in antiquity, and is proposition I.47 in [[Euclid's Elements|Euclid's ''Elements'']]: "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."

===Area===
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 <math>T</math> is
:<math>T=\tfrac{1}{2}ab</math>

where <math>a</math> and <math>b</math> are the legs of the triangle.

If the [[Incircle and excircles of a triangle|incircle]] is tangent to the hypotenuse <math>AB</math> at point <math>P,</math> then letting the [[semi-perimeter]] be <math>s = \tfrac12(a+b+c),</math> we have  <math>|PA| = s-a</math> and <math>|PB| = s-b,</math> and the area is given by
:<math>T= |PA| \cdot |PB| = (s-a)(s-b).</math>

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>

===Altitudes===
[[Image:Teorema.png|thumb|right|Altitude {{mvar|f}} of a right triangle]]<!--Better diagram needed here.-->
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; these are both [[Similarity (geometry)|similar]] to the original, and therefore similar to each other. From this:
* The altitude to the hypotenuse is the [[geometric mean]] ([[Ratio#Euclid's definitions|mean proportional]]) of the two segments of the hypotenuse.<ref name=Posamentier/>{{rp|243}}
* Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
In equations,
:<math>f^2=de,</math> (this is sometimes known as the [[right triangle altitude theorem]])
:<math>b^2=ce,</math>
:<math>a^2=cd</math>

where <math>a,b,c,d,e,f</math> are as shown in the diagram.<ref>Wentworth p. 156</ref> Thus
:<math>f=\frac{ab}{c}.</math>

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>
:<math>\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{f^2}.</math>

For solutions of this equation in integer values of <math>a,b,c,f,</math> see [[Integer triangle#Pythagorean triangles with integer altitude from the hypotenuse|here]].

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.

===Inradius and circumradius===
The radius of the [[incircle]] of a right triangle with legs <math>a</math> and <math>b</math> and hypotenuse <math>c</math> is
:<math>r = \frac{a+b-c}{2} = \frac{ab}{a+b+c}.</math>

The radius of the [[circumcircle]] is half the length of the hypotenuse,
:<math>R = \frac{c}{2}.</math>

Thus the sum of the circumradius and the inradius is half the sum of the legs:<ref name="Crux">''Inequalities proposed in "[[Crux Mathematicorum]]"'', [http://www.imomath.com/othercomp/Journ/ineq.pdf].</ref>

:<math>R+r = \frac{a+b}{2}.</math>

One of the legs can be expressed in terms of the inradius and the other leg as
:<math>a=\frac{2r(b-r)}{b-2r}.</math>