Editing Right triangle (section)

===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.