How to prove a formula for the midpoint of a line between two vectors

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how to prove that OP1 + 0,5 * P1P2 = OC?

C is the midpoint of P1P2.

Thank you

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Show that $OC-OP_1=OP_2-OC$, which is perhaps the definition of midpoint.

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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: $$\vec{OP_1}+\vec{CP_1}=\vec{OC}$$