  We uncover a new type of magic-angle phenomena when an AA-stacked graphene
bilayer is twisted relative to another graphene system with band touching. In
the simplest case this constitutes a trilayer system formed by an AA-stacked
bilayer twisted relative to a single layer of graphene. We find multiple
anisotropic Dirac cones coexisting in such twisted multilayer structures at
certain angles, which we call "Dirac magic." We trace the origin of Dirac magic
angles to the geometric structure of the twisted AA-bilayer Dirac cones
relative to the other band-touching spectrum in the moir\'e reciprocal lattice.
The anisotropy of the Dirac cones and a concomitant cascade of saddle points
induce a series of topological Lifshitz transitions that can be tuned by the
twist angle and perpendicular electric field. We discuss the possibility of
direct observation of Dirac magic as well as its consequences for the
correlated states of electrons in this moir\'e system.
