Vectors in Space · Hard problem

A triangle has edge vectors $\mathbf a$, $\mathbf b$, $\mathbf c$, each pointing the way around the triangle from one vertex to the next, so that $\mathbf a+\mathbf b+\mathbf c = \mathbf 0$. Their lengths are $\lVert\mathbf a\rVert=3$, $\lVert\mathbf b\rVert=4$, $\lVert\mathbf c\rVert=5$. Find $\mathbf a\cdot\mathbf b+\mathbf b\cdot\mathbf c+\mathbf c\cdot\mathbf a$.