Lami's Theorem

Statement. If three coplanar, concurrent forces $\vec F_1,\vec F_2,\vec F_3$ keep a point in equilibrium, then the magnitude of each force is proportional to the sine of the angle between the other two, with the same constant of proportionality for all three:

Here $\alpha$ is the angle between $\vec F_2$ and $\vec F_3$ (i.e. the angle "opposite" $\vec F_1$), and similarly for $\beta$ and $\gamma$.

Worked application (the swing/chain problem). A baby on a swing hangs at rest from two identical chains. Three forces act at the point where the chains meet the seat: gravity $mg$ (down) and the two chain tensions $T$ (each at some angle $\theta$ to the vertical). Applying Lami's theorem (or, equivalently, resolving vertically) gives

When $\theta=0°$ (chains exactly vertical), this correctly reduces to $T=mg/2$ — each chain simply supports half the weight. …