Solve mx=xa+xb/2my=ya+yb/2

$$mx = \frac{ xa+xb }{ 2 } my = \frac{ ya+yb }{ 2 }$$

Evaluate (complex solution)

$$mx=\frac{mxy\left(a+b\right)}{2}\text{ and }\frac{mxy\left(a+b\right)}{2}=\frac{y\left(a+b\right)}{2}$$

Solve for x

$$\left\{\begin{matrix}x=0\text{, }&y=0\text{ or }a=-b\\x\in \mathrm{R}\text{, }&\left(a=-b\text{ or }y=0\right)\text{ and }m=0\\x=\frac{1}{m}\text{, }&a=-b+\frac{2}{y}\text{ and }y\neq 0\text{ and }m\neq 0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=\frac{2}{a+b}\text{, }&a\neq -b\text{ and }m=\frac{1}{x}\text{ and }x\neq 0\\y=0\text{, }&x=0\text{ or }m=0\\y\in \mathrm{R}\text{, }&\left(x=0\text{ or }m=0\right)\text{ and }a=-b\end{matrix}\right.$$