Single crystal magnetic studies combined with a theoretical analysis show that cancellation of the magnetic moments in the trinuclear Dy3+ cluster [Dy{3}(mu{3}-OH)2L3Cl(H2O){5}]Cl{3}, resulting in a nonmagnetic ground doublet, originates from the noncollinearity of the single-ion easy axes of magnetization of the Dy3+ ions that lie in the plane of the triangle at 120 degrees one from each other. This gives rise to a peculiar chiral nature of the ground nonmagnetic doublet and to slow relaxation of the magnetization with abrupt accelerations at the crossings of the discrete energy levels.