Look at the picture above, graphically/geometrically, the green line is equal to the sum of the blue and red lines. But I am trying to think of an algebraical proof. Let's say the lengths of the side are $a$, $b$ and $c$, respectively, and the angles between the sides and the line onto which the sides are projected are $\alpha$, $\beta$ and $\gamma$, respectively.
i.e. the question is: How to prove that $$a\cos{\alpha} + b\cos{\beta} = c\cos{\gamma}$$? I am always willing to be enlightened. Thank you very much.