**Basic Results involving parallelogram**

The points $A$ , $B$, $C$, $D$ have position vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$ respectively relative to an origin $O$. If $P$ divides $AB$ in the ratio $1:2$ and $Q$ divides $CD$ in the ratio $1:2$, obtain an expression for the position vector of $X$, where $X$ is the mid-point of $PQ$.

If $ABCD$ is a parallelogram, show that $X$ is the point in which diagonals $AC$ and $BD$ intersect.

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