For discussion sake, let’s say a body experiences three and only three forces, F1, F2 and F3.
Remember that if a body is in static equilibrium, the summation of moments about any point must be zero.
Let’s exercise our freedom to choose our pivot point wherever we like. Let’s choose point X, where the lines of action of F1 and F2 intersect. Can you see that if the line of action of F3 does not pass through point X, then the summation of moments about X will not be zero!?! Because F1 and F2 both contribute zero moment about X, F3 must also contribute zero moment. The only way for F3 to contribute zero moment about X is if its line of action also passes through X.
Thus with this simple logic, we arrive at this rather cute rule that for a three-force system at static equilibrium, (the lines of action of) all three forces must intersect at one common point. 🙂
P.S. This rule is not applicable if all three forces are parallel to one another. Why?
P.S. This rule is also not applicable if a body experiences 4 or more forces. Why?
P.S. If you prefer to hear me explain, click here.