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.
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 3-parameter 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 $a=\ell k^2$, $b=\ell km$, $c=\ell(m^2-k^2)$; 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, $2k^2 < m^2 < 4k^2$.