If $\vec a, \vec b, \vec c$ and $\vec d$ are the position vectors of points A, B, C, D such that no three of them are collinear and $\vec a+\vec c=\vec b+\vec d$, then ABCD is a

parallelogram

We have,

$\vec a+\vec c=\vec b+\vec d$

$⇒\vec b-\vec a=\vec c-\vec d$ and $\vec d-\vec a=\vec c-\vec b$

$⇒\vec{AB}=\vec{DC}$ and $\vec{AD}=\vec{BC}$

$⇒ABCD$ is a parallelogram.