In non-premixed turbulent combustion the reactive zone is localized at the stoichiometric surfaces of the mixture and may be locally approximated by a diffusion flame. Experiments and numerical simulations reveal a characteristic structure at the edge of such a two-dimensional diffusion flame. This ‘triple flame’ or ‘edge flame’ consists of a curved flame front followed by a trailing edge that constitutes the body of the diffusion flame. Triple flames are also observed at the edge of a lifted laminar diffusion flame near the exit of burners. The speed of propagation of the triple flame determines such important properties as the rate of increase of the flame surface in non-premixed combustion and the lift-off distance in lifted flames at burners. This paper presents an approximate theory of triple flames based on an approximation of the flame shape by a parabolic profile, for large activation energy and low but finite heat release. The parabolic flame path approximation is a heuristic approximation motivated by physical considerations and is independent of the large activation energy and low heat release assumptions which are incorporated through asymptotic expansions. Therefore, what is presented here is not a truly asymptotic theory of triple flames, but an asymptotic solution of a model problem in which the flame shape is assumed parabolic. Only the symmetrical flame is considered and Lewis numbers are taken to be unity. The principal results are analytical formulas for the speed and curvature of triple flames as a function of the upstream mixture fraction gradient in the limit of infinitesimal heat release as well as small but finite heat release. For given chemistry, the solution provides a complete description of the triple flame in terms of the upstream mixture fraction gradient. The theory is validated by comparison with numerical simulation of the primitive equations.