The brachistochrone (shortest-time) curve is the path connecting two points that enables the shortest travel time. This work explores the "brachistochrone path" of fire spread connecting two points at the same altitude and with a fixed path length. The starting and ending points are connected by both thermally thin fuels (thin wires) and thermally thick fuels (PMMA bars). Flame-spread paths of triangular, rectangular, and circular shapes with different heights and inclinations are explored. Results show that having a local maximum flame-spread rate does not result in the shortest overall travel time. For thin-wire paths, the fastest overall-path fire spread occurs, when the upward spread path is vertical, and the path height reaches a maximum, as demonstrated by the theoretical analysis. Differently, for thick PMMA-bar paths, the brachistochrone condition occurs when the path length of the vertical upward spread reaches the maximum, because the upward spread is about ten times faster than the downward spread. This study extends the conventional problem of the fastest fire spread to the shortest-time problem of the whole fire path, and it may help optimize the fuel distribution inside the built environment and estimate available safe egress time in building and wildland fires.