1. Given:

   • $C$ is the midpoint of $AB$.
   • $P$ is any point outside $AB$.

2. Use position vectors: Let the position vectors of points $A,B,C,P$ be $\vec a, \vec b, \vec c, \vec p$ respectively.

   Since $C$ is the midpoint of $AB$, $\vec c = \frac{\vec a + \vec b}{2}.$

3. Write the required vectors: $\overrightarrow{PA} = \vec a - \vec p,$ $\overrightarrow{PB} = \vec b - \vec p,$ $\overrightarrow{PC} = \vec c - \vec p.$

4. Add $\overrightarrow{PA}$ and $\overrightarrow{PB}$: $\overrightarrow{PA} + \overrightarrow{PB} = (\vec a - \vec p) + (\vec b - \vec p)$ $= \vec a + \vec b - 2\vec p.$

5. Now compute $2\overrightarrow{PC}$: $2\overrightarrow{PC} = 2(\vec c - \vec p) = 2\vec c - 2\vec p.$

   Using $\vec c = \dfrac{\vec a + \vec b}{2}$, $2\vec c - 2\vec p = (\vec a + \vec b) - 2\vec p.$

   Hence, $2\overrightarrow{PC} = \vec a + \vec b - 2\vec p.$

6. Compare both expressions: $\overrightarrow{PA} + \overrightarrow{PB} = \vec a + \vec b - 2\vec p = 2\overrightarrow{PC}.$

   Therefore, $\boxed{\overrightarrow{PA} + \overrightarrow{PB} = 2\overrightarrow{PC}}.$

7. Check options:

   • A: Correct
   • B: Incorrect, missing factor $2$
   • C: Incorrect, because the vectors are not necessarily $\vec 0$
   • D: Incorrect

So the correct option is A.