Three force theorem

A system of three forces in equilibrium must be coplanar and either be concurrent or be parallel.

Proof: Assume that the two of the forces F₁ and F₂ intersect at a point A. The resultant of these two forces is given by,

R = F₁+F₂

The third force is equal and opposite of this force. Hence, it must be in the plane of F₁ and F₂. If it does not pass through the point A, it will produce a couple. Hence, if two forces intersect at a point, the third force is in the same plane and passes through the same point.

Now,assume that the forces do not intersect. Assume that forces F₁ passes through point A. If B and C are some points on F₂ and F₃,respectively, then taking moment about A,

If M be, some other point on the line of action of force F_1, then taking moment about M,