If are in the same set then are in the same set, the relation is symmetric.
Thus far the relation is no different than most equivalence relations.
But proving it is transitive requires a bit more.
You need to have proved this theorem: If two connected subsets have a point in common then their union is connected.
So if are in the same connected subset and are in the same connected subset then the union is connected having in common. Thus are in the same connected subset.