Applied Mechanics and Graphic Statics
If a set of given forces are such that their free vectors build a closed polygon, then

The resultant force is zero but resultant couple is not zero
The resultant force is zero but resultant couple may not be zero
The resultant force and resultant couple are always zero
The resultant force and resultant couple both may not be zero

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Applied Mechanics and Graphic Statics
Lami's theorem states that

Three forces acting at a point are always in equilibrium
If three forces acting on a point can be represented in magnitude and direction by the sides of a triangle, the point will be in the state of equilibrium
Three coplanar forces acting at a point will be in equilibrium if each force is inversely proportional to the sine of the angle between the other two
Three coplanar forces acting at a point will be in equilibrium, if each force is proportional to the sine of the angle between the other two

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