Given a convex polygon and an environment consisting of polygonal obstacles, we find the largest similar copy of that does not intersect any of the obstacles. Allowing translation, rotation, and change-of-size, our method combines a new notion of Delaunay triangulation for points and edges with the well-known functions based on Davenport-Schinzel sequences producing an almost quadratic algorithm for the problem. Namely, if is a convex -gon and if has corners and edges then we can find the placement of the largest similar copy of in the environment in time log), where \lambda_{4} is one of the almost-linear functions related to Davenport-Schinzel sequences. If the environment consists only of points then we can find the placement of the largest similar copy of in time log ).