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.

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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 $$ 4 \overrightarrow{E F} $$.

$$
\boxed{\text{Resultant} = 4\, \overrightarrow{E F}}
$$
The resultant of the forces is $$ 4 \overrightarrow{E F} $$.

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