In order to study whether equipower property can decide the global property of a space，the possibleregion in which equipower points of the unit circle might exist is presented in a non- strictly convex normed plane viageometric method． By applying known results concerning transitivity of the norm，it is proved that if the unit circleof a Minkowski plane admits an equipower point other than the origin as well as a big point，then the space is a Hil-bert space．