Two forces \( \overrightarrow{A B} \) and \( \overrightarrow{A D} \) are acting at vertex \( A \) of a quadrilateral \( A B C D \) two forces \( \overrightarrow{C B} \) and \( \overrightarrow{C D} \) at \( C \). Prove their resultant is given by \( 4 \overrightarrow{E F} \), wh E and \( F \) are the midpoints of \( \overrightarrow{A C} \) anc \( \overrightarrow{B D} \), respectively.

To prove that the resultant of the forces \( \overrightarrow{A B} \), \( \overrightarrow{A D} \), \( \overrightarrow{C B} \), and \( \overrightarrow{C D} \) is equal to \( 4 \overrightarrow{E F} \), where \( E \) and \( F \) are the midpoints of \( \overrightarrow{A C} \) and \( \overrightarrow{B D} \) respectively, we'll use vector addition and properties of midpoints. ### Step 1: Define Position Vectors Let's assign position vectors to the points: - Let \( \mathbf{a} \), \( \mathbf{b} \), \( \mathbf{c} \), and \( \mathbf{d} \) be the position vectors of points \( A \), \( B \), \( C \), and \( D \) respectively. ### Step 2: Express the Given Forces in Terms of Position Vectors The vectors representing the forces are: \[ \overrightarrow{A B} = \mathbf{b} - \mathbf{a} \] \[ \overrightarrow{A D} = \mathbf{d} - \mathbf{a} \] \[ \overrightarrow{C B} = \mathbf{b} - \mathbf{c} \] \[ \overrightarrow{C D} = \mathbf{d} - \mathbf{c} \] ### Step 3: Calculate the Resultant Force The resultant \( \mathbf{R} \) is the vector sum of these forces: \[ \mathbf{R} = (\mathbf{b} - \mathbf{a}) + (\mathbf{d} - \mathbf{a}) + (\mathbf{b} - \mathbf{c}) + (\mathbf{d} - \mathbf{c}) \] \[ \mathbf{R} = 2\mathbf{b} + 2\mathbf{d} - 2\mathbf{a} - 2\mathbf{c} \] \[ \mathbf{R} = 2(\mathbf{b} + \mathbf{d} - \mathbf{a} - \mathbf{c}) \] ### Step 4: Determine the Vector \( \overrightarrow{E F} \) First, find the position vectors of midpoints \( E \) and \( F \): \[ \mathbf{e} = \frac{\mathbf{a} + \mathbf{c}}{2} \quad \text{(Midpoint of \( A C \))} \] \[ \mathbf{f} = \frac{\mathbf{b} + \mathbf{d}}{2} \quad \text{(Midpoint of \( B D \))} \] Now, the vector \( \overrightarrow{E F} \) is: \[ \overrightarrow{E F} = \mathbf{f} - \mathbf{e} = \frac{\mathbf{b} + \mathbf{d}}{2} - \frac{\mathbf{a} + \mathbf{c}}{2} \] \[ \overrightarrow{E F} = \frac{\mathbf{b} + \mathbf{d} - \mathbf{a} - \mathbf{c}}{2} \] ### Step 5: Relate the Resultant to \( \overrightarrow{E F} \) Multiplying \( \overrightarrow{E F} \) by 4: \[ 4 \overrightarrow{E F} = 4 \times \frac{\mathbf{b} + \mathbf{d} - \mathbf{a} - \mathbf{c}}{2} = 2(\mathbf{b} + \mathbf{d} - \mathbf{a} - \mathbf{c}) = \mathbf{R} \] ### Conclusion Thus, the resultant of the forces is indeed equal to \( 4 \overrightarrow{E F} \). \[ \boxed{\text{Resultant} = 4\, \overrightarrow{E F}} \]