Proving half turns

Question

I am proving that if M is the midpoint of $\overline{AB}$ then the translation represented by $\overrightarrow{AB}$ equals $H_M H_A$ and $H_B H_M$

So we know $H_B H_A$ equals a translation that is two times the distance of $AB$. Can I just say how $H_M H_A$ will be half the translation that $H_B H_A$ was? Then we know that $H_M H_A + H_B H_M = AB$

Answer 1

You already know that $H_B H_A = T_{2 \vec{AB}}$? Then you also know $H_M H_A = T_{2 \vec{AM}} = T_{\vec{AB}}$, similarly for $H_B H_M$.