Basic geometric question on areas spanned by vectors and the triangle inequality

Question

I came across this diagram in here:

I'm probably missing something obvious, but this seems to contradict the triangle inequality, and I would imagine to be correct only when $\vec c =\vec 0.$

Answer 1

if all the vectors $\underline{a}$, $\underline{b}$ and $\underline{c}$, and therefore also $\underline{b}+\underline{c}$ are co-planar then it is easy to see that you can add the areas of parallelograms $CEFD$ and $ACDB$, then subtract area $ACE$ and add area $DBF$ then you do indeed have area $AEFB$.

You may be misreading the diagram as a parallelipiped.