In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem

$$\frac{A}{\sin{\alpha}}=\frac{B}{\sin{\beta}}=\frac{C}{\sin{\gamma}}.\tag{1}$$

where $A$, $B$ and $C$ are the magnitudes of the three coplanar, concurrent and non-collinear vectors, $V_A$, $V_B$, $V_C$, which keep the object in static equilibrium, and $α$, $β$ and $γ$ are the angles directly opposite to the vectors. See an illustration here.

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.

Generalization. If four coplanar, concurrent and non-collinear vectors act upon an object, and the object remains in static equilibrium, then

$$\frac{AB+CD}{\sin{\alpha}}=\frac{AD+BC}{\sin{\beta}}=\frac{AB+CD}{\sin{\gamma}}=\frac{AD+BC}{\sin{\delta}}.\tag{2}$$

where $A$, $B$, $C$ and $D$ are the magnitudes of the four vectors and $\alpha$, $\beta$, $\gamma$ and $\delta$ are the angles between the vectors such that $\alpha+\gamma=\pi=\beta+\delta$ (see figure above).

Proof. The proof relies on our generalization of the sine law, which is not hard to prove. Indeed, consider the cyclic quadrilateral formed by the four vectors in such a manner that the head of one touches the tail of another (see figure below) and denote $\Delta$ its area. Keep in mind that $\alpha+\gamma=\pi=\beta+\delta$. Since $\Delta=(AD+BC)\sin\alpha=(AB+CD)\sin\beta$ the relation in $(2)$ follows.

The relation in $(2)$ generalizes $(1)$ in the sense that if one of the vectors vanishes, the relation we obtain is that of Lami's theorem. Consider the relation $$\frac{AB+CD}{\sin\alpha}=\frac{AD+BC}{\sin\beta}.\tag{3}$$

Suppose that $C=0$, then $(3)$ reduces to

$$\frac{B}{\sin\alpha}=\frac{D}{\sin\beta},$$

which is Lami's theorem.

Questions: Is Lami's theorem used in the design of cranes as the image below suggest? If so, could the generalization presented be used in the design of more efficient cranes given that we have 4 vectors instead of 3?