# 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 .

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

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

and one cathetus is the harmonic mean

 $\displaystyle h\;=\;\frac{2uv}{u\!+\!v}$ (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 [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.

## 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