There are two known lower bounds for μ(P, Q) in an EpG, called β1 and β2, see for example [3]. In [4], μ=β1 was studied for the case of triangular EGQs and, in [3], μ=β2 was considered for EpGs in general. Here we extend this to the case μ=β1 for EpGs in general, including non-triangular EGQs, and we give a number of characterizations. For instance a triangular EpG with μ=β1 locally is an EGQ, an extended dual net or a semibiplane; if t>2α−1, then an EpGα(s, t) with μ=β1 locally is an EGQ. In general we have only partial results for t≤2α−1.