Lami's theorem

Remarks

Proof idea: placing the three vectors head to tail gives a closed triangle, since they sum to zero. The interior angle of this triangle opposite the side $F_A$ is $\pi - \alpha$, and similarly for the others. The law of sines and $\sin(\pi - \alpha) = \sin\alpha$ give the result. The hypotheses ensure that no two of the forces are parallel, so all three angles lie strictly between $0$ and $\pi$ and the sines are nonzero.

Example: a weight of $W$ newtons hangs from two strings that each make an angle of $60^\circ$ with the vertical. The angle between the two strings is $120^\circ$ and each string makes $180^\circ - 60^\circ = 120^\circ$ with the downward weight. So $W / \sin 120^\circ = T / \sin 120^\circ$, and each string tension $T$ equals $W$.

Going deeper

For a student

Lami's theorem is the law of sines applied to the triangle of forces. It is a special case of the equilibrium conditions of statics for concurrent forces, where the torque condition is automatic. For more than three forces, or forces not meeting at one point, one uses the full equilibrium equations instead.