If `P` is a point and `A B C D` is a quadrilateral and ` vec A P+ vec P B+ vec P D= vec P C` , show that `A B C D` is a parallelogram.

Explanation

To show that quadrilateral \( ABCD \) is a parallelogram given that \( \vec{AP} + \vec{PB} + \vec{PD} = \vec{PC} \), we can follow these steps:

### Step 1: Define the position vectors
Let \( P \) be the origin. We denote the position vectors of points \( A, B, C, D \) as \( \vec{A}, \vec{B}, \vec{C}, \vec{D} \) respectively. Thus, we have:
- \( \vec{P} = \vec{0} \) (since \( P \) is the origin)
- \( \vec{AP} = \vec{P} - \vec{A} = \vec{0} - \vec{A} = -\vec{A} \)
- \( \vec{PB} = \vec{B} - \vec{P} = \vec{B} - \vec{0} = \vec{B} \)
- \( \vec{PD} = \vec{D} - \vec{P} = \vec{D} - \vec{0} = \vec{D} \)
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