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# 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

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

## Mathematics Subject Classification

11D09*no label found*51M05

*no label found*

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