The structures of m-benzyne and its fluorinated derivative, tetrafluoro-m-benzyne, were investigated using coupled cluster methods including triple excitations [CCSD(T) and CCSDT], different reference wave functions (spin-restricted Hartree-Fock, spin-unrestricted Hartree-Fock, and Brueckner), and different basis sets [6-31G(d,p) and correlation-consistent valence triple-zeta (cc-pVTZ)]. The inclusion of triple excitations in conjunction with d- and f-type polarization functions is paramount to correctly describe through-bond delocalization of the monocyclic form. At the highest level of theory, the C1-C3 distance of the minimum energy form of m-benzyne is 2.0 A and the profile of the potential energy surface along the C1-C3 distance is that of an asymmetric, single well, in agreement with previous density-functional theory and coupled cluster studies. In addition, the calculated CCSD(T) fundamental frequencies are in excellent agreement with the measured infrared frequencies, thus confirming the monocyclic form of m-benzyne. For tetrafluoro-m-benzyne, however, the increased eclipsing strain between the ring-external C-X bonds stabilizes the bicyclo[3.1.0]hexatriene form: the C1-C3 distance is calculated at the CCSD(T)/cc-pVTZ level to be approximately 1.75 A, which is in the range of elongated CC bonds. Computed harmonic vibrational frequencies compare reasonably well with the experimental neon-matrix difference spectrum and provide further evidence for the existence of a bicyclic form.