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 differentintegers, the smallest of them being at least 3.

Any Pythagorean triangle has the property that the hypotenuse isthe 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 $u$, $v$; theother cathetus is simply $|u-v|$.

If there is given the value of $c$ as the length of thehypotenuse and a compatible value  $h$ as the length of onecathetus, the pair of equations (1) and (2) does not determinethe numbers $u$ and $v$ uniquely (cf. the Proposition 4 in theentry 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 alsoin the PlanetMath article contraharmonic means and Pythagoreanhypotenuses.  The contraharmonic and the harmonic mean of twointegers are simultaneously integers (seethis article (http://planetmath.org/IntegerHarmonicMeans)).  The aboveclaim concerning the catheti of the Pythagorean triangle isevident 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 bythe equations (1) and (2) are

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

as one instantly sees by substituting them into the equations.