Consider triangles with angles alpha, beta, gamma such that gamma = 2 alpha, and sides (a,b,c) are integers. I want to parametrize such triangles by a single integer or rational variable.

$\begingroup$ Have you tried to use the law of cosines? $\endgroup$ – Somos Feb 16 at 2:34

1$\begingroup$ There are two degrees of freedom here. Not possible to use only a single variable to parametrize the triangles. $\endgroup$ – Somos Feb 16 at 3:10

$\begingroup$ I meant, up to similarity. $\endgroup$ – Michael Beeson Feb 16 at 4:59

$\begingroup$ I guessed as much. Please edit your question accordingly. $\endgroup$ – Somos Feb 16 at 5:00
See Konstantine Zelator, Integral Triangles with one angle twice another, and with the bisector(of the double angle) also of integral length. The abstract says, "In Result 2 in Section 5, we offer 3parameter formulas that describe the entire family of integral triangles $ABC$ with $\angle A=2\angle B$." But it seems it's actually Result 1.
Result 1: The entire family of integral triangles $ABC$, with angle at $B$ being twice the angle at $A$; can be parametrically described (in terms of three integer parameters) as follows:
Sidelengths $a=\ell k^2$, $b=\ell km$, $c=\ell(m^2k^2)$; where $\ell,m,k$ are positive integers such that $k$ and $m$ are relatively prime and with either, $k^2 < m^2 < 2k^2$ or alternatively, $2k^2 < m^2 < 4k^2$.

$\begingroup$ See also R. S. Luthar, Integersided triangles with one angle twice another, College Mathematics Journal, January, 1984, available at maa.org/sites/default/files/0746834215798.di020710.02p0187t.pdf The question was also given as a coding challenge in 2015: codewars.com/kata/… $\endgroup$ – Gerry Myerson Feb 16 at 4:17

$\begingroup$ The question asks for a single variable parametrization. $\endgroup$ – Somos Feb 16 at 5:03

$\begingroup$ k/m can be regarded as the single rational parameter I asked for. $\endgroup$ – Michael Beeson Feb 16 at 5:28