Equilibrium of Concurrent Forces

Several forces are said to be concurrent when the lines along which they act all pass through one single common point, even though the forces themselves may point in entirely different directions. A point acted on by a number of concurrent forces is in equilibrium exactly when the vector sum of all those forces is zero -- equivalently, when the forces, drawn one after another (head to tail, in any order) as vectors, form a closed polygon with no gap left between the start of the first vector and the end of the last.

For the particular case of exactly three concurrent forces $P$, $Q$ and $R$ keeping a point in equilibrium, this closed-polygon condition becomes a closed triangle, and it can be restated as Lami's theorem: each force is proportional to the sine of the angle between the other two forces. Labelling $\alpha$ as the angle between $Q$ and $R$ (i.e. opposite $P$ in the force triangle), $\beta$ as the angle between $R$ and $P$ (opposite $Q$), and $\gamma$ as the angle between $P$ and $Q$ (opposite $R$), with $\alpha + \beta + \gamma = 360^\circ$,

Lami's theorem is especially convenient because it lets any one of the three forces be found directly from the other two and the angles between them, without first having to resolve every force into horizontal and vertical components -- though resolving into components (setting $\sum F_x = 0$ and $\sum F_y = 0$ separately) always works too, and gives exactly the same answer, since Lami's theorem is nothing more than a restated, purely geometric form of that same vector equilibrium condition. …