||We present a new quadtree-based decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum weight Steiner triangulation (MWST) to within a constant factor. An approximate MWST is good for ray shooting in the average case as defined by Aronov and Fortune. The untruncated decomposition also yields an approximate MWST. Moreover, we show that this triangulation supports query-sensitive ray shooting as defined by Mitchell, Mount, and Suri. Hence, there exists a Steiner triangulation that is simultaneously good for ray shooting in the query-sensitive sense and in the average case.