Drawing a graph symmetrically enables an understanding of the entire graph to be built up from that of smaller subgraphs. This paper discusses symmetric drawings of planar graphs. More specifically, we discuss planar geometric automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar geometric automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. In this paper, we present a polynomial time algorithm for finding planar geometric automorphisms of graphs.