`A B C D` are four points in a plane and `Q` is the point of intersection of the lines joining the mid-points of `A B` and `C D ; B C` and `A Ddot` Show that ` vec P A+ vec P B+ vec P C+ vec P D=4 vec P Q ,` where `P` is any point.

Explanation

To prove that \( \vec{PA} + \vec{PB} + \vec{PC} + \vec{PD} = 4 \vec{PQ} \), we will use the properties of vectors and the concept of midpoints.

### Step 1: Define the points and midpoints
Let the position vectors of points \( A, B, C, D \) be represented as \( \vec{A}, \vec{B}, \vec{C}, \vec{D} \) respectively. The midpoints of segments \( AB \) and \( CD \) can be defined as:
- Midpoint \( M_1 \) of \( AB \):
\[ \vec{M_1} = \frac{\vec{A} + \vec{B}}{2} \]
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