proof of cosines law

Let $a$, $b$, $c$ be the sides of a triangle and $\alpha$, $\beta$, $\gamma$ its angles, respectively.  By the projection formula, one may write the equalities

$\{\begin{array}{cc}a=b\cos \gamma +c\cos \beta \hfill & \\ b=c\cos \alpha +a\cos \gamma \hfill & \\ c=a\cos \beta +b\cos \alpha .\hfill & \end{array}$

Multiplying the equalities by $a$, $-b$ and $-c$, respectively, they read

$\{\begin{array}{cc}{a}^2=ab\cos \gamma +ca\cos \beta \hfill & \\ -b^2=-bc\cos \alpha -ab\cos \gamma \hfill & \\ -c^2=-ca\cos \beta -bc\cos \alpha .\hfill & \end{array}$

Addition of these yields the sum equation

$$a^2-b^2-c^2=-2bc\cos \alpha ,$$

i.e.

$$a^2=b^2+c^2-2bc\cos \alpha ,$$

which is the cosines law.