A generalization of Lami's theorem for 4 forces

 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

$$\frac{F_1}{\sin{\alpha}}=\frac{F_2}{\sin{\beta}}=\frac{F_3}{\sin{\gamma}}.\tag{1}$$

where $F_1$, $F_2$ and $F_3$ are the magnitudes of the three coplanar, concurrent and non-collinear vectors which keep the object in static equilibrium, and $\alpha$, $\beta$ and $\gamma$ are the angles directly opposite to the vectors (see Figure 1).

Figure 1. $\alpha$, $\beta$ and $\gamma$ are the angles directly opposite to the vectors $F_1$, $F_2$ and $F_3$

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy. The proof of Lami's theorem is essentially based on the law of sines.

On the Internet there are hundreds of static equilibrium problems where they apply Lami's theorem to a three-force system, see for instance Dubey - Engineering Mechanics: Statics and Dynamics, section 3.10. Although Dubey's book is recent (2013) there is not a single equilibrium problem based on a four-force system. Coincidentally, the author of this note has come across questions on the Internet questioning the possibility of applying Lami's theorem for more than three forces. In this note we give a generalization of Lami's theorem for four forces.

Theorem 1 (Generalization). If four coplanar, concurrent and non-collinear forces act upon an object, and the object remains in static equilibrium, then

Figure 2. An example of the situation described in the theorem 1. 

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

where $A$, $B$, $C$ and $D$ are the magnitudes of the four vectors and $\alpha'$, $\beta'$, $\gamma'$ and $\delta'$ are the angles between them (see Figure 2).

Proof. Consider the quadrilateral formed by the four vectors in such a manner that the head of one touches the tail of another (see Figure 3) and denote $\Delta$ its area. If $\alpha$, $\beta$, $\gamma$ and $\delta$ are the interior angles of the quadrilateral, then its area can be written as

Figure 3. Notice that if $\alpha$, $\beta$, $\gamma$ and $\delta$ are the interior angles of the quadrilateral formed by the four vectors, then $\alpha$ and $\alpha'$ are supplementary and similarly for $\beta'$, $\gamma'$ and $\delta'$

$$\Delta=\frac12AD\sin{\alpha}+\frac12BC\sin{\gamma}=\frac12AB\sin{\beta}+\frac12CD\sin{\delta}$$ 

and as $\sin{\alpha'}=\sin{(\pi-\alpha)}=\sin{\alpha}$, and similarly for $\beta'$, $\gamma'$ and $\delta'$, the relation in $(2)$ follows.

$\square$

Theorem 1 is a generalization in the sense that if one of the vectors vanishes, the relation we obtain is that of Lami's theorem. Indeed, for instance suppose $C=0$, then the relation $(2)$ reduces to

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

which is Lami's theorem.

Remark. A generalization of Lami's theorem is given by H. Shekhar. However, this generalization is different since it only considers cyclic polygons with an odd number of sides.