Pythagorean triangle

The side lengths of any right triangle satisfy the equation of the Pythagorean theorem, but if they are integers then the triangle is a Pythagorean triangle.

The side lengths are said to form a Pythagorean triple.  They are always different integers, the smallest of them being at least 3.

Any Pythagorean triangle has the property that the hypotenuse is the contraharmonic mean

$c={\displaystyle \frac{u^2+v^2}{u+v}}$ (1)

and one cathetus is the harmonic mean

$h={\displaystyle \frac{2uv}{u+v}}$ (2)

of a certain pair of distinct positive integers $u$, $v$; the other cathetus is simply $|u-v|$.

If there is given the value of $c$ as the length of the hypotenuse and a compatible value  $h$ as the length of one cathetus, the pair of equations (1) and (2) does not determine the numbers $u$ and $v$ uniquely (cf. the Proposition 4 in the entry integer contraharmonic means).  For example, if  $c=61$  and  $h=11$, then the equations give for  $(u,v)$  either  $(6,\mathrm{\hspace{0.17em}66})$  or  $(55,\mathrm{\hspace{0.17em}66})$.

As for the hypotenuse and (1), the proof is found in [1] and also in the PlanetMath article contraharmonic means and Pythagorean hypotenuses.  The contraharmonic and the harmonic mean of two integers are simultaneously integers (see this article (http://planetmath.org/IntegerHarmonicMeans)).  The above claim concerning the catheti of the Pythagorean triangle is evident from the identity

$${\left(\frac{2uv}{u+v}\right)}^2+{\left|u-v\right|}^2={\left(\frac{u^2+v^2}{u+v}\right)}^2.$$

If the catheti of a Pythagorean triangle are $a$ and $b$, then the values of the parameters $u$ and $v$ determined by the equations (1) and (2) are

${\displaystyle \frac{c+b\pm a}{2}}$ (3)

as one instantly sees by substituting them into the equations.

Title	Pythagorean triangle
Canonical name	PythagoreanTriangle
Date of creation	2013-11-23 11:53:13
Last modified on	2013-11-23 11:53:13
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Result
Classification	msc 11D09
Classification	msc 51M05