Applied Mechanics and Graphic Statics
A body of weight ‘w’ placed on an inclined plane is acted upon by a force ‘P’ parallel to the plane which causes the body just to move up the plane. If the angle of inclination of the plane is ‘θ’ and angle of friction is ‘φ’, the minimum value of ‘P’, is

w sin (φ - θ)/cos φ
w sin (θ - φ)/cos φ
w sinθ cos(θ - φ)/sin φ
w cos (θ + φ)/cos φ

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Applied Mechanics and Graphic Statics
The Law of Polygon of Forces states that

If forces acting on a point can be represented in magnitude and direction by the sides of a polygon in order, the forces are in equilibrium
If a polygon representing the forces acting at point in a body is closed, the forces are in equilibrium
If forces acting on a point can be represented of a polygon taken in order, their sides of a polygon taken in order, their resultant will be represented in magnitude and direction by the closing side
If forces acting on a point can be represented in magnitude and direction by the sides of a polygon taken in order, then the resultant of the forces will be represented in magnitude and direction by t

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