Abstract: EVH black holes are Extremal black holes with Vanishing Horizon area, where vanishing of horizon area is a result of having a vanishing one-cycle on the horizon. We prove three theorems regarding near horizon geometry of EVH black hole solutions to generic Einstein gravity theories in diverse dimensions: (1) The near horizon geometry of any EVH black hole has a three dimensional maximally symmetric subspace. (2) If the energy momentum tensor of the theory satisfies strong energy condition either this 3d part is an AdS3, or the solution is a direct product of a locally 3d flat space and a d−3 dimensional part. (3) These results extend to the near horizon geometry of near-EVH black holes, for which the AdS3 part is replaced with BTZ geometry. We discuss that our theorems apply to Einstein-Maxwell-dilaton theories, and to gauged or ungauged supergravity theories with U(1) Maxwell fields.