Accurate traveltime computation in complex geometries is crucial for seismological applications such as traveltime tomography and migration. However, eikonal solvers relying on finite-differences encounter the source singularity, leading to numerical errors that propagate throughout the computational domain and compromise traveltime accuracy. Solving the factored eikonal equation has proven effective in addressing this singularity, but it has not yet been applied to triangular meshes, which are significant for modeling complex geological structures and irregular topographies. To address this challenge, we propose a new coordinate-transformation-based fast sweeping method (FSMCT) designed for triangular meshes. FSMCT maps each triangular element in the physical domain into a canonical reference triangle through coordinate transformation, and constructs an upwind finite-difference scheme on the reference triangle to solve the eikonal equation. The framework enables a straight forward extension to the multiplicatively/additively factored eikonal equations, overcoming the long-standing limitation that the conventional fast sweeping method (FSM) cannot solve the factored eikonal equation on triangular meshes, thereby effectively mitigating source singularity. Furthermore, FSMCT solves the eikonal equation on the reference triangle, while existing FSM typically require further subdivision of obtuse triangles. Numerical simulations demonstrate that FSMCT accurately solves the factored eikonal equation on triangular meshes, substantially reducing the errors caused by source singularity and achieving significantly higher accuracy than the conventional FSM.