I am rather confused. Suppose $V$ is a finite dimensional vector space and $A,B,C$ are (non-trivial) subspaces of $V$ such that $V=A\oplus B=A\oplus C=B\oplus C$ and it is said that there is a subspace of dimension 2 of $V$ whose intersection with any one of $A,B,C$ is a one dimensional subspace.
My confusion: Now since $V$ is a direct sum of these pairs of these subspaces this means that the pairwise intersection of $A,B,C$ has to be $\{0\}$. But then $A\oplus B=A\oplus C$ means that $B,C$ must be the same space, which is clearly wrong. What is going on here?