proof of Apollonius theorem

Let $b=\overline{CA}$, $a=\overline{BC}$, $c=\overline{AB}$, and $m=\overline{AM}$. Let $\mathrm{\angle }CMA=\theta$, so that $\mathrm{\angle }BMA=\pi -\theta$.

By the law of cosines, $b^2=m^2+\frac{a^2}{4}-am\cos \theta$ and $c^2=m^2+\frac{a^2}{4}-am\cos (\pi -\theta )=m^2+\frac{a^2}{4}+am\cos \theta$, and adding gives

$$b^2+c^2=2m^2+\frac{a^2}{2}.$$

QED