Applications for generalized Lami's theorem

Question

Hello fellow physics enthusiasts! I recently came across with a generalization of Lami's theorem for four coplanar, concurrent and non-collinear forces in static equilibrium. I was wondering if anyone could suggest some potential applications or real-world scenarios where Theorem 1 could be useful? Additionally, I'm curious if this theorem can simplify or facilitate calculations compared to other methods. Any insights or thoughts would be appreciated.

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

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

Answer 1

Lami's theorem (For Three Vectors) is just another way of manipulating vector algebra with the help of the Sine Law.

We could generalize this theorem to multiple vectors (In your case, Forces).

We can apply this theorem to easily solve problems in theoretical physics but when it comes to real-world applications, mainly such theorems are used to solve modeled real-world problems without the use of computers and fancy vector algebraic calculators (Quite common in 19th and 20th century engineering works)! Perhaps, while building the Golden Gate Bridge, engineers could have used the Lami's theorem to calculate the resultant stress and strain across those suspension cables :)

I've seen household ladders having a maximum load imprinted on them. We could calculate this value from the angles made by the steps with the rafts and the rigidity of the metal from which the ladder is made. I could not come up with other real world applications :) !

But take this – Such theorems are just used in theoretical physics which deals with idealized objects and phenomena. Perhaps, some physics enthusiast would trifle his time by calculating the resultants of some real world physical quantities with such theorems :)

Otherwise, these days, all professional physics problems use computers which have much more advanced theorems muddled up into them as algorithms!