Lami's theorem

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

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

where A, B and C are the magnitudes of three coplanar, concurrent and non-collinear forces, which keep the object in static equilibrium, and
α, β and γ are the angles directly opposite to the forces A, B and C respectively.

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

Proof of Lami's Theorem

Suppose there are three coplanar, concurrent and non-collinear forces, which keeps the object in static equilibrium. By the triangle law, we can re-construct the diagram as follow:

By the law of sines,

$\frac{A}{\sin (\pi - \alpha)}=\frac{B}{\sin (\pi - \beta)}=\frac{C}{\sin (\pi - \gamma)}$
$\Rightarrow \frac{A}{\sin \alpha}=\frac{B}{\sin \beta}=\frac{C}{\sin \gamma}$