# parametrize triangles meeting certain conditions

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.

• Have you tried to use the law of cosines? – Somos Feb 16 at 2:34
• There are two degrees of freedom here. Not possible to use only a single variable to parametrize the triangles. – Somos Feb 16 at 3:10
• I meant, up to similarity. – Michael Beeson Feb 16 at 4:59
• I guessed as much. Please edit your question accordingly. – 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, $$2k^2 < m^2 < 4k^2$$.