We investigate finite linear spaces whose incidence graph satisfies a condition of the following type: given six integers i, j, k, l, m, n, there is an integer c such that for any triple (u, v, w) of vertices of where u is a point and d(u, v) = i,d(u, w) = j, d(w, u) = k, there are exactly c vertices of which are at distance l from u, mfrom v, and n from w. The most interesting of these conditions lead essentially to semi-affine planes and generalized projective spaces. In this paper, we prove the equivalence between some of these conditions.