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 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, $2k^2 < m^2 < 4k^2$.


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