If we can find a sequence of vectors that come back to where they started when placed head to tail, then we will have found a combination that adds up to zero. Hence, we try to form a triangle with the vectors $\bfa, \bfb, \bfc$.

To form a triangle, we start by trying to combine $\bf{a}$ and $\bf{b}$ in a way that produces a horizontal vector. It appears that two copies of $\mathbf{b}$ will work:

It appears that $\mathbf{a} + 2 \mathbf{b}$ is about half as long as $\bf{c}$ and points in the opposite direction.