In this paper we resolve the following problem: Given a simple polygon , what is the maximum-area polygon that is axially symmetric and is contained by ? We propose an algorithm for answering this question, analyze the algorithm’s complexity, and describe our implementation of it (for convex polygons). The algorithm is based on building and investigating a planar map, each cell of which corresponds to a different configuration of the inscribed polygon. We prove that the complexity of the map is , where is the complexity of . For a convex polygon the complexity, in the worst case, is .