Calculate the bisection vector OM

Question

Calculate the bisection vector OM, by using the vectors that form an angle with it, $OA_1$ and $OA_2$

I was thinking of using the Angle bisector theorem, but i can't seem to get that end formula look.

Answer 1

$$\begin{cases} \vec{OM}=\vec{OA_1}+\vec{A_1M}\\ \vec{OM}=\vec{OA_2}+\vec{A_2M} \end{cases}$$ $$\begin{cases} \vec{OM}=\vec{OA_1}+k\vec{MA_2}\\ \vec{OM}=\vec{OA_2}-\vec{MA_2} \end{cases}$$ $$\begin{cases} \vec{OM}=\vec{OA_1}+k\vec{MA_2}\\ k\vec{OM}=k\vec{OA_2}-k\vec{MA_2} \end{cases}$$

Now , add two equations to obtain :

$$(1+k)\vec{OM}=\vec{OA_1}+k\vec{OA_2}$$

from which we have :

$$\vec{OM}=\frac{\vec{OA_1}+k\vec{OA_2}}{1+k}$$