The merging of two lines of fire is a relatively common occurrence in landscape fire events. For example, it can arise through the coalescence of two wildfires or when a prescribed fire meets a wildfire as part of suppression efforts. When two fires approach one another, the effects of convective and radiative heat transfer are compounded and high rates of spread can arise as a result. This is particularly the case when two oblique lines of fire meet at some acute angle - the point of intersection on the newly merged fire can advance rapidly. This case was investigated recently by Viegas et al. (2012), who devised a simple analytical model to emulate the effects of energy concentration between the two merging fire lines. In this paper, we present a more geometric approach by considering the evolution of the merged fire as the flow of a plane curve with a normal speed that depends on the curvature of the fire front.