If $C$ is the mid-point of the line segment $\mathbf{A B}$ and $P$ is any point outside the line $A B$, then

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Aicharya · Accepted Answer

$\mathbf{P A}+\mathbf{A C}=\mathbf{P C}$ ...(i) $\mathbf{P B}+\mathbf{B C}=\mathbf{P C}$ ...(ii) On adding Eqs. (i) and (ii), $\overrightarrow{\mathbf{P A}}+\overrightarrow{\mathbf{P B}}+\overrightarrow{\mathbf{A C}}+\overrightarrow{\mathbf{B C}}=2 \mathbf{P C}$ $\Rightarrow \mathbf{P A}+\mathbf{P B}=2 \mathbf{P C} \quad[\because \mathbf{B C}=-\mathbf{A C}]$

If $C$ is the mid-point of the line segment $\mathbf{A B}$ and $P$ is any point outside the line $A B$, then

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