This paper deals with the determination of skeletons and conditional bisectors in discrete binary images using the Euclidean metrics. The algorithm proceeds in two steps: first, the Centers of the Euclidean Maximal Discs (CMD) included in the set to skeletonize are characterized and robustly identified. Second, a firefront propagation is simulated starting from the set boundaries, in which pixels which are not centers of maximal discs and are not crucial to homotopy preservation are removed. Not only is the resulting algorithm fast and accurate, it allows the computation of a vast variety of skeletons. Furthermore, it can be extended to provide conditional bisectors of any angular parameter (theta) . This leads to the introduction of a new morphological transformation, the bisector function, which synthesizes the information contained in all the (theta) -conditional bisectors. The interest of all these skeleton-like transformations is illustrated on the segmentation of binary images of glass fibers.