Show that $OC-OP_1=OP_2-OC$, which is perhaps the definition of midpoint.
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Bumbble Comm
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As you said, being $C$ is the mid point of $\vec{P_1P_2}$, you have:
$$\vec{P_1C}=\vec{CP_1}$$
Now, by the definition of the sum between vectors, follows that:
$$\vec{OP_1}+\vec{CP_1}=\vec{OC}$$
Show that $OC-OP_1=OP_2-OC$, which is perhaps the definition of midpoint.