contraharmonic means and Pythagorean hypotenuses

One can see that all values of $c$ in the table of the parent entry (http://planetmath.org/IntegerContraharmonicMeans) are hypotenuses in a right triangle with integer sides (http://planetmath.org/Triangle).  E.g., 41 is the contraharmonic mean of 5 and 45;  $9^2+{40}^2={\mathrm{\hspace{0.33em}41}}^2$.

Theorem.  Any integer contraharmonic mean of two different positive integers is the hypotenuse of a Pythagorean triple.  Conversely, any hypotenuse of a Pythagorean triple is contraharmonic mean of two different positive integers.

Proof.  $1^{\circ }.$  Let the integer $c$ be the contraharmonic mean

$$c=\frac{u^2+v^2}{u+v}$$

of the positive integers $u$ and $v$ with  $u>v$.  Then  $u+v\mid u^2+v^2={(u+v)}^2-2uv$, whence

$$u+v\mid \mathrm{\hspace{0.17em}2}uv,$$

and we have the positive integers

$$a=:u-v=\frac{u^2-v^2}{u+v},b=:\frac{2uv}{u+v}$$

satisfying

$$a^2+b^2=\frac{{(u^2-v^2)}^2+{(2uv)}^2}{{(u+v)}^2}=\frac{u^4-2u^2{v}^2+v^4+4u^2{v}^2}{{(u+v)}^2}=\frac{u^4+2u^2{v}^2+v^4}{{(u+v)}^2}=\frac{{(u^2+v^2)}^2}{{(u+v)}^2}=c^2.$$

$2^{\circ }.$  Suppose that $c$ is the hypotenuse of the Pythagorean triple  $(a,b,c)$,  whence $c^2=a^2+b^2$.  Let us consider the rational numbers

$u=:{\displaystyle \frac{c+b+a}{2}},v=:{\displaystyle \frac{c+b-a}{2}}.$(1)

If the triple is primitive (http://planetmath.org/PythagoreanTriple), then two of the integers $a,b,c$ are odd and one of them is even; if not, then similarly or all of $a,b,c$ are even.  Therefore, $c+b\pm a$ are always even and accordingly $u$ and $v$ positive integers.  We see also that  $u+v=c+b$.  Now we obtain

$u^2+v^2$$={\displaystyle \frac{c^2+b^2+a^2+2ab+2bc+2ca+c^2+b^2+a^2-2ab+2bc-2ca}{4}}$

$={\displaystyle \frac{2c^2+2(a^2+b^2)+4bc}{4}}={\displaystyle \frac{4c^2+4bc}{4}}=c(c+b)$

$=c(u+v).$

Thus, $c$ is the contraharmonic mean ${\displaystyle \frac{u^2+v^2}{u+v}}$ of the different integers $u$ and $v$. (N.B.:  When the values of $a$ and $b$ in (1) are changed, another value of $v$ is obtained.  Cf. the Proposition 4 in theparent entry (http://planetmath.org/IntegerContraharmonicMeans).)