Fat triangles determine linearly many holes

The authors show that for every fixed δ > 0 the following holds: if F is a union of n triangles, all of whose angles are at least δ, then the complement of F has O(n) connected components and the boundary of F consists of O(n log log n) straight segments (where the constants of proportionality depend on δ). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges.