Triangle Law of Forces:
If three forces acting at a point are represented in magnitude and direction by the sides of a triangle, taken in order, then they will be in equilibrium.
Proof: Let the forces P, Q, and R acting at a point O be represented in magnitude and direction by the sides AB, BC, and CA respectively of △ABC.
| The parallelogram ABCD is completed. ∴ BC || AD and BC = AD  Now by parallelogram law of forces  Thus the resultant of P and Q is represented by   i.e., the resultant of the forces P, Q, and R is zero. Hence the forces P, Q, and R are in equilibrium. |
Notes: (1) From the triangle law of forces we can say that if the forces P, Q, and R are proportional to the sides AB, BC, and CA respectively, of △ABC, even then the forces will be in equilibrium.
(2) When three forces are in equilibrium, each is equal and opposite to the resultant of the other two. For example, the resultant of forces AB and BC are equal and opposite to CA, i.e., their resultant is represented by AC.
Converse of the Triangle Law of Forces:
If three forces acting at a point are in equilibrium, then they can be represented in magnitude and direction by the sides of any triangle that is drawn so as to have its sides respectively parallel to the direction of the forces.
Proof: Let the forces P, Q, and R acting at O be in equilibrium. Taking a suitable scale and measuring OA and OC represent the forces P and Q respectively.
The parallelogram OABC is completed. OB is drawn. Since the forces are in equilibrium, each is equal and opposite to the resultant of the other two. In this case, R must be equal and opposite to the resultant of P and Q and must, therefore, be represented by BO. Also, AB is equal and parallel to OC. Thus, the forces P, Q, and R are parallel and proportional to the sides OA, AB, and BO of △OAB. This proves the theorem.
Notes: (1) If three forces acting at a point are in equilibrium, then they can be represented by any triangle similar to △OAB.
(2) If three forces acting at a point are such that the sum of the magnitudes of any two of them is less than the magnitude of the third force, then they can never be in equilibrium, since in that case, they can never be represented by the sides of a triangle.
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