Cosine Inequality

Take three vectors $\mathbf a, \mathbf b,$ and $\mathbf c$. We have $$(\mathbf a + \mathbf b + \mathbf c) \cdot (\mathbf a + \mathbf b + \mathbf c) \ge 0,$$ so $$\lVert \mathbf a\rVert^2 + \lVert \mathbf b\rVert^2 + \lVert \mathbf c\rVert^2 + 2(\mathbf a \cdot \mathbf b) + 2(\mathbf a \cdot \mathbf c) + 2(\mathbf b \cdot \mathbf c) \ge 0.$$

Now take $a,b,c$ to be the lengths, and $A,B,C$ to be the angles opposite the vectors. Using the fact that $\mathbf a \cdot \mathbf b = \lVert \mathbf a\rVert \cdot \lVert \mathbf b\rVert \cos C$, and similarly for the other two dot products, we get the desired inequality.