Choosing a sheltered middle path

We say that point x∈R2 is sheltered by a continuum S⊂R2 if x does not belong to the unbounded component of R2\S. Suppose that points a and b are the endpoints of each of three arcs A0, A1 and A2 contained in R2. We prove that there is an arc B⊂A0∪A1∪A2 with its endpoints a and b such that each point of B is sheltered by the union of each two of the arcs A0, A1 and A2.