  Recently, Hu and Xi have exhibited derived equivalent endomorphism rings
arising from (relative) almost split sequences as well as AR-triangles in
triangulated categories. We present a broader class of triangles (in algebraic
triangulated categories) for which the endomorphism rings of different terms
are derived equivalent. We then study applications involving 0-Calabi-Yau
triangulated categories. In particular, applying our results in the category of
perfect complexes over a symmetric algebra gives a nice way of producing pairs
of derived equivalent symmetric algebras. Included in the examples we work out
are some of the algebras of dihedral type with two or three simple modules. We
also apply our results to stable categories of Cohen-Macaulay modules over
odd-dimensional Gorenstein hypersurfaces having an isolated singularity.
