Choosing a sheltered middle path

We say that point x ∈ R 2 is sheltered by a continuum S ⊂ R 2 if x does not belong to the unbounded component of R 2 \ S . Suppose that points a and b are the endpoints of each of three arcs A 0 , A 1 and A 2 contained in R 2 . We prove that there is an arc B ⊂ A 0 ∪ A 1 ∪ A 2 with its endpoints a and b such that each point of B is sheltered by the union of each two of the arcs A 0 , A 1 and A 2 .