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Statement

A triangle with angles $(\alpha,\beta,\gamma)$ satisfying $3\alpha+2\beta=\pi$ has rational side ratios if and only if $\sin(\alpha/2)\in\mathbb Q$.

Source and scope. Beeson–Laczkovich–Zhang, arXiv:2604.03609v3, Proposition 9, pp. 3–4. Complete rewritten proof.

Proof

Let $a,b,c$ be opposite $\alpha,\beta,\gamma$. The angle relation gives $\beta=(\pi-3\alpha)/2$ and $\gamma=(\pi+\alpha)/2$. The sine rule and the triple-angle identity therefore give

$$\frac ac=\frac{\sin\alpha}{\cos(\alpha/2)}=2\sin(\alpha/2),\qquad \frac bc=\frac{\cos(3\alpha/2)}{\cos(\alpha/2)} =1-4\sin^2(\alpha/2).$$

If $\sin(\alpha/2)$ is rational, both ratios are rational. Conversely, rationality of $a/c$ gives rationality of $\sin(\alpha/2)$. Thus either condition is equivalent to commensurability of all three sides.

Dependencies. Sine rule and elementary trigonometric identities.

Bears on. Problem 633.