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If $C$ is the mid point of $A B$ and $P$ is any point outside $A B$, then
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2871 Upvotes
Verified Answer
The correct answer is:
$\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}=2 \overrightarrow{\mathrm{PC}}$
$\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}=2 \overrightarrow{\mathrm{PC}}$
$$
\begin{aligned}
& \overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{CP}}=0 \\
& \overline{\mathrm{PB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CP}}=0
\end{aligned}
$$
Adding, we get
$$
\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{BC}}+2 \overrightarrow{\mathrm{CP}}=0
$$
Since $\overrightarrow{A C}=-\overrightarrow{B C}$
$$
\begin{aligned}
& \& \overrightarrow{\mathrm{CP}}=-\overrightarrow{\mathrm{PC}} \\
& \Rightarrow \overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}-2 \overrightarrow{\mathrm{PC}}=0 .
\end{aligned}
$$
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