# Pythagorean triangle

 $\displaystyle c\;=\;\frac{u^{2}\!+\!v^{2}}{u\!+\!v}$ (1)
 $\displaystyle h\;=\;\frac{2uv}{u\!+\!v}$ (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  $(6,\,66)$  or  $(55,\,66)$.

As for the hypotenuse and (1), the proof is found in  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.

## References

• 1 J. Pahikkala: “On contraharmonic mean and Pythagorean triples”.  – Elemente der Mathematik 65:2 (2010).
Title Pythagorean triangle PythagoreanTriangle 2013-11-23 11:53:13 2013-11-23 11:53:13 pahio (2872) pahio (2872) 15 pahio (2872) Result msc 11D09 msc 51M05