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
$\frac{c+b\pm a}{2}$ | (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 |
---|---|
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 |