Consider ΔABC and $Δ\, A_1\, B_1\, C_1$ in such a way that $\vec{AB} = \vec{A_1B_1}$ and $M, N, M_1, N_1$ be the mid-points of $AB, BC, A_1B_1$ and $B_1C_1$ respectively. Then,

$\vec{MM_1}=\vec{BB_1}$

We have,

$\vec{AB} = \vec{A_1B_1}$

$⇒\vec b- \vec a = \vec{b_1}-\vec{a_1}$

$⇒\vec b- \vec{b_1}=\vec a -\vec{a_1}⇒\vec{B_1B}=\vec{A_1A}⇒\vec{AA_1}=\vec{BB_1}$

Now,

$\vec{NN_1}=\frac{\vec{b_1}+\vec{c_1}}{2}-\frac{\vec{b}+\vec{c}}{2}=\frac{\vec{b_1}+\vec{c_1}-\vec{b}-\vec{c}}{2}$

$⇒2\vec{NN_1}=\vec{BB_1}+\vec{CC_1}$

and,

$\vec{MM_1}=\frac{\vec{b_1}+\vec b+\vec{a_1}+\vec a}{2}$

$⇒2\vec{MM_1}=\vec{BB_1}+\vec{AA_1}$

$⇒2\vec{MM_1}=2\vec{BB_1}$  $[∵\vec{AA_1}=\vec{BB_1}]$

$⇒\vec{MM_1}=\vec{BB_1}=\vec{AA_1}$