This paper is devoted to the rigorous study of the low temperature properties of the two-dimensional weakly interacting Hubbard model on the honeycomb lattice in which the renormalized chemical potential $μ$ has been fixed such that the Fermi surface consists of a set of exact triangles. Using renormalization group analysis around the Fermi surface, we prove that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. The main result is proved in two steps. First we prove that the perturbation series for Schwinger functions as well as the self-energy function have non-zero radius of convergence when the temperature $T$ is above an exponentially small value, namely ${T_0\sim \exp{(-C|λ|^{-1/2})}}$. Then we prove the necessary lower bound for second derivatives of self-energy w.r.t. the external momentum and achieve the proof.