proof of Apollonius theoremLet b=CA¯, a=BC¯, c=AB¯, and m=AM¯. Let ∠CMA=θ, so that ∠BMA=π-θ.By the law of cosines, b2=m2+a24-amcosθ and c2=m2+a24-amcos(π-θ)=m2+a24+amcosθ, and adding givesb2+c2=2m2+a22.QED