The problem of a flame generated by three coaxial flows is solved by extending the Burke-Schumann methodology to include a third stream. The solution is particularly relevant to flame synthesis wherein multiple tubes are often employed either to introduce inert as a diffusion barrier or to introduce more than two reactants. The general problem is solved where the inner and outer tubes contain reactants and the middle tube contains either an inert or a third reactant. Relevant examples are considered and the results show that the triaxial Burke-Schumann flame can be substantially more complicated than the traditional Burke-Schumann flame. When the middle flow is inert the flame temperature is no longer constant but increases axially, reaching a maximum at the flame centerline. At the exit the flame does not sit on the tube exit but instead resides between the inner and outer tubes, resulting in an effective barrier for particle build-up on the burner rim. For the case of a third reactant in the middle flow, synthesis chemistry where the inner reaction is endothermic and the outer reaction is exothermic is considered. In addition to showing the flame temperature and flame shape, the results identify conditions wherein reaction is not possible due to insufficient heat transfer from the outer flame to support the inner flame reaction.