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A set A subset of V is resolved by S if all its elements are resolved by S. S is a resolving set in Gamma if it resolves V. The metric dimension of Gamma is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.\n\nWe show that the metric dimension of the incidence graph of a finite projective plane of order g >= 23 is 4g - 4, and describe all resolving sets of that size. Let tau(2) denote the size of the smallest double blocking set in PG(2, q), the Desarguesian projective plane of order q. We prove that for a semi-resolving set S in the in graph of PG(2, q), vertical bar S vertical bar >= min{2g + q/4 - 3, tau(2) - 2} holds. In particular, if q >= 9 is a square, then the smallest semi-resolving set in PG(2, q) has size 2g + 2 root q. 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